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Properties

Label 10-6012e5-1.1-c1e5-0-1
Degree $10$
Conductor $7.854\times 10^{18}$
Sign $1$
Analytic cond. $2.54964\times 10^{8}$
Root an. cond. $6.92864$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·5-s − 2·7-s + 5·11-s − 8·13-s + 7·17-s + 2·19-s + 13·23-s + 13·25-s + 11·29-s − 12·31-s − 14·35-s − 7·37-s + 12·41-s + 19·47-s − 20·49-s + 21·53-s + 35·55-s + 7·59-s − 6·61-s − 56·65-s + 10·67-s + 35·71-s − 8·73-s − 10·77-s + 11·83-s + 49·85-s + 32·89-s + ⋯
L(s)  = 1  + 3.13·5-s − 0.755·7-s + 1.50·11-s − 2.21·13-s + 1.69·17-s + 0.458·19-s + 2.71·23-s + 13/5·25-s + 2.04·29-s − 2.15·31-s − 2.36·35-s − 1.15·37-s + 1.87·41-s + 2.77·47-s − 2.85·49-s + 2.88·53-s + 4.71·55-s + 0.911·59-s − 0.768·61-s − 6.94·65-s + 1.22·67-s + 4.15·71-s − 0.936·73-s − 1.13·77-s + 1.20·83-s + 5.31·85-s + 3.39·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 3^{10} \cdot 167^{5}\)
Sign: $1$
Analytic conductor: \(2.54964\times 10^{8}\)
Root analytic conductor: \(6.92864\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{10} \cdot 3^{10} \cdot 167^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(28.84556217\)
\(L(\frac12)\) \(\approx\) \(28.84556217\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167$C_1$ \( ( 1 - T )^{5} \)
good5$C_2 \wr S_5$ \( 1 - 7 T + 36 T^{2} - 134 T^{3} + 394 T^{4} - 981 T^{5} + 394 p T^{6} - 134 p^{2} T^{7} + 36 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 2 T + 24 T^{2} + 41 T^{3} + 286 T^{4} + 405 T^{5} + 286 p T^{6} + 41 p^{2} T^{7} + 24 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 5 T + 36 T^{2} - 136 T^{3} + 628 T^{4} - 2025 T^{5} + 628 p T^{6} - 136 p^{2} T^{7} + 36 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 8 T + 64 T^{2} + 292 T^{3} + 1433 T^{4} + 4915 T^{5} + 1433 p T^{6} + 292 p^{2} T^{7} + 64 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 7 T + 79 T^{2} - 402 T^{3} + 2557 T^{4} - 9625 T^{5} + 2557 p T^{6} - 402 p^{2} T^{7} + 79 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 2 T + 50 T^{2} - 86 T^{3} + 1389 T^{4} - 2425 T^{5} + 1389 p T^{6} - 86 p^{2} T^{7} + 50 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 13 T + 144 T^{2} - 1052 T^{3} + 6778 T^{4} - 34395 T^{5} + 6778 p T^{6} - 1052 p^{2} T^{7} + 144 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 11 T + 166 T^{2} - 1174 T^{3} + 9958 T^{4} - 49617 T^{5} + 9958 p T^{6} - 1174 p^{2} T^{7} + 166 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 12 T + 153 T^{2} + 1291 T^{3} + 9382 T^{4} + 57399 T^{5} + 9382 p T^{6} + 1291 p^{2} T^{7} + 153 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 7 T + 178 T^{2} + 920 T^{3} + 12752 T^{4} + 48883 T^{5} + 12752 p T^{6} + 920 p^{2} T^{7} + 178 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 12 T + 193 T^{2} - 1363 T^{3} + 13060 T^{4} - 69271 T^{5} + 13060 p T^{6} - 1363 p^{2} T^{7} + 193 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 107 T^{2} - 197 T^{3} + 6754 T^{4} - 10885 T^{5} + 6754 p T^{6} - 197 p^{2} T^{7} + 107 p^{3} T^{8} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 19 T + 168 T^{2} - 1313 T^{3} + 12955 T^{4} - 108573 T^{5} + 12955 p T^{6} - 1313 p^{2} T^{7} + 168 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 21 T + 350 T^{2} - 4071 T^{3} + 40465 T^{4} - 314151 T^{5} + 40465 p T^{6} - 4071 p^{2} T^{7} + 350 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 7 T + 196 T^{2} - 789 T^{3} + 15775 T^{4} - 43795 T^{5} + 15775 p T^{6} - 789 p^{2} T^{7} + 196 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 6 T + 158 T^{2} + 1262 T^{3} + 14037 T^{4} + 105179 T^{5} + 14037 p T^{6} + 1262 p^{2} T^{7} + 158 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 10 T + 142 T^{2} - 236 T^{3} - 205 T^{4} + 59767 T^{5} - 205 p T^{6} - 236 p^{2} T^{7} + 142 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 35 T + 763 T^{2} - 11235 T^{3} + 130765 T^{4} - 1205501 T^{5} + 130765 p T^{6} - 11235 p^{2} T^{7} + 763 p^{3} T^{8} - 35 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 8 T + 312 T^{2} + 2092 T^{3} + 42635 T^{4} + 220341 T^{5} + 42635 p T^{6} + 2092 p^{2} T^{7} + 312 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 197 T^{2} + 805 T^{3} + 17746 T^{4} + 119621 T^{5} + 17746 p T^{6} + 805 p^{2} T^{7} + 197 p^{3} T^{8} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 11 T + 326 T^{2} - 2557 T^{3} + 45301 T^{4} - 272481 T^{5} + 45301 p T^{6} - 2557 p^{2} T^{7} + 326 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 32 T + 606 T^{2} - 7762 T^{3} + 881 p T^{4} - 741105 T^{5} + 881 p^{2} T^{6} - 7762 p^{2} T^{7} + 606 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 11 T + 505 T^{2} - 4153 T^{3} + 99605 T^{4} - 598793 T^{5} + 99605 p T^{6} - 4153 p^{2} T^{7} + 505 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.90490666378709603930894278170, −4.56559163654757471439683260914, −4.48508890222424990811994495222, −4.42759416258627250006456211627, −4.18560860694675874696339415684, −3.80654779492185920198237424852, −3.63942655137263077135661495446, −3.62510106029962839840631464629, −3.49050390440047606152257497226, −3.42126363551032204969124764996, −3.09376458736996657602643132728, −2.79497242727545626472397138540, −2.71028360404887738067007013675, −2.58105470431620863279746074866, −2.42492235863041341943804309631, −2.20414976932895680863978180055, −2.06116010626006796329874322884, −1.81165033501310444938528462250, −1.71479057497070911595844381146, −1.70789573585979855324745199154, −1.03559393555351406408400121008, −0.973076977190740593462311448105, −0.947662512915993581495484390034, −0.60523250832563126221853186895, −0.42439600878059316446815568226, 0.42439600878059316446815568226, 0.60523250832563126221853186895, 0.947662512915993581495484390034, 0.973076977190740593462311448105, 1.03559393555351406408400121008, 1.70789573585979855324745199154, 1.71479057497070911595844381146, 1.81165033501310444938528462250, 2.06116010626006796329874322884, 2.20414976932895680863978180055, 2.42492235863041341943804309631, 2.58105470431620863279746074866, 2.71028360404887738067007013675, 2.79497242727545626472397138540, 3.09376458736996657602643132728, 3.42126363551032204969124764996, 3.49050390440047606152257497226, 3.62510106029962839840631464629, 3.63942655137263077135661495446, 3.80654779492185920198237424852, 4.18560860694675874696339415684, 4.42759416258627250006456211627, 4.48508890222424990811994495222, 4.56559163654757471439683260914, 4.90490666378709603930894278170

Graph of the $Z$-function along the critical line