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Properties

Label 10-671e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.360\times 10^{14}$
Sign $-1$
Analytic cond. $4415.68$
Root an. cond. $2.31472$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·4-s − 2·5-s − 7-s + 8·8-s − 9·9-s + 4·10-s + 5·11-s − 10·13-s + 2·14-s + 5·16-s − 3·17-s + 18·18-s − 13·19-s + 6·20-s − 10·22-s − 18·25-s + 20·26-s − 3·27-s + 3·28-s − 7·29-s − 13·31-s − 15·32-s + 6·34-s + 2·35-s + 27·36-s − 6·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 3/2·4-s − 0.894·5-s − 0.377·7-s + 2.82·8-s − 3·9-s + 1.26·10-s + 1.50·11-s − 2.77·13-s + 0.534·14-s + 5/4·16-s − 0.727·17-s + 4.24·18-s − 2.98·19-s + 1.34·20-s − 2.13·22-s − 3.59·25-s + 3.92·26-s − 0.577·27-s + 0.566·28-s − 1.29·29-s − 2.33·31-s − 2.65·32-s + 1.02·34-s + 0.338·35-s + 9/2·36-s − 0.986·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{5} \cdot 61^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{5} \cdot 61^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(11^{5} \cdot 61^{5}\)
Sign: $-1$
Analytic conductor: \(4415.68\)
Root analytic conductor: \(2.31472\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 11^{5} \cdot 61^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad11$C_1$ \( ( 1 - T )^{5} \)
61$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p T + 7 T^{2} + 3 p^{2} T^{3} + 3 p^{3} T^{4} + 33 T^{5} + 3 p^{4} T^{6} + 3 p^{4} T^{7} + 7 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + p^{2} T^{2} + p T^{3} + 38 T^{4} + 17 T^{5} + 38 p T^{6} + p^{3} T^{7} + p^{5} T^{8} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 2 T + 22 T^{2} + 34 T^{3} + 41 p T^{4} + 241 T^{5} + 41 p^{2} T^{6} + 34 p^{2} T^{7} + 22 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + T + 3 p T^{2} + 27 T^{3} + 233 T^{4} + 257 T^{5} + 233 p T^{6} + 27 p^{2} T^{7} + 3 p^{4} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 10 T + 99 T^{2} + 561 T^{3} + 3014 T^{4} + 11183 T^{5} + 3014 p T^{6} + 561 p^{2} T^{7} + 99 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 3 T + 84 T^{2} + 199 T^{3} + 167 p T^{4} + 5033 T^{5} + 167 p^{2} T^{6} + 199 p^{2} T^{7} + 84 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 13 T + 144 T^{2} + 1024 T^{3} + 6356 T^{4} + 29521 T^{5} + 6356 p T^{6} + 1024 p^{2} T^{7} + 144 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 80 T^{2} + 101 T^{3} + 2826 T^{4} + 4587 T^{5} + 2826 p T^{6} + 101 p^{2} T^{7} + 80 p^{3} T^{8} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 7 T + 78 T^{2} + 6 T^{3} - 100 T^{4} - 14311 T^{5} - 100 p T^{6} + 6 p^{2} T^{7} + 78 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 13 T + 163 T^{2} + 1437 T^{3} + 10393 T^{4} + 64421 T^{5} + 10393 p T^{6} + 1437 p^{2} T^{7} + 163 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 6 T + 153 T^{2} + 749 T^{3} + 10446 T^{4} + 39095 T^{5} + 10446 p T^{6} + 749 p^{2} T^{7} + 153 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 9 T + 206 T^{2} + 1354 T^{3} + 16734 T^{4} + 80727 T^{5} + 16734 p T^{6} + 1354 p^{2} T^{7} + 206 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 2 T + 149 T^{2} - 97 T^{3} + 10204 T^{4} - 1983 T^{5} + 10204 p T^{6} - 97 p^{2} T^{7} + 149 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 3 T + 213 T^{2} + 457 T^{3} + 18859 T^{4} + 29687 T^{5} + 18859 p T^{6} + 457 p^{2} T^{7} + 213 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 3 T + 135 T^{2} - 607 T^{3} + 10497 T^{4} - 45141 T^{5} + 10497 p T^{6} - 607 p^{2} T^{7} + 135 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 14 T + 322 T^{2} + 2995 T^{3} + 38600 T^{4} + 256597 T^{5} + 38600 p T^{6} + 2995 p^{2} T^{7} + 322 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 5 T + 249 T^{2} + 870 T^{3} + 27449 T^{4} + 72939 T^{5} + 27449 p T^{6} + 870 p^{2} T^{7} + 249 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 3 T + 282 T^{2} - 1111 T^{3} + 34521 T^{4} - 127665 T^{5} + 34521 p T^{6} - 1111 p^{2} T^{7} + 282 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 4 T + 247 T^{2} + 1069 T^{3} + 28266 T^{4} + 113873 T^{5} + 28266 p T^{6} + 1069 p^{2} T^{7} + 247 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 27 T + 654 T^{2} + 9608 T^{3} + 125752 T^{4} + 1182335 T^{5} + 125752 p T^{6} + 9608 p^{2} T^{7} + 654 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 3 T + 272 T^{2} + 278 T^{3} + 34402 T^{4} + 8271 T^{5} + 34402 p T^{6} + 278 p^{2} T^{7} + 272 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 12 T + 370 T^{2} + 3531 T^{3} + 58932 T^{4} + 442409 T^{5} + 58932 p T^{6} + 3531 p^{2} T^{7} + 370 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 5 T + 258 T^{2} + 578 T^{3} + 37892 T^{4} + 80189 T^{5} + 37892 p T^{6} + 578 p^{2} T^{7} + 258 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.96447000545605885432459726530, −6.95027774940935982446546029834, −6.33310456187507958406643792996, −6.30498621919867198686755985530, −6.11355043638659871035896916060, −6.01614542834553048334039879602, −5.70911749078672148659568727425, −5.42049599438066676918451385701, −5.38544076306636054804861513856, −5.32004334725503082438528115541, −4.80359661336207276368356103687, −4.56481798372561365305760113005, −4.52454673920528179282044836093, −4.32772884101188457736166321598, −4.10910739803291499959014413306, −3.72142371110374203940543256383, −3.65475666888024064720906916621, −3.43065950650081936329356244241, −3.36477418145787138409837655988, −2.82813204279153685511136258919, −2.50537167906142119342436934326, −2.32100931804778924790119235032, −1.99449765795089920808176596516, −1.72764744079302892728895887532, −1.48701628137430090011069279546, 0, 0, 0, 0, 0, 1.48701628137430090011069279546, 1.72764744079302892728895887532, 1.99449765795089920808176596516, 2.32100931804778924790119235032, 2.50537167906142119342436934326, 2.82813204279153685511136258919, 3.36477418145787138409837655988, 3.43065950650081936329356244241, 3.65475666888024064720906916621, 3.72142371110374203940543256383, 4.10910739803291499959014413306, 4.32772884101188457736166321598, 4.52454673920528179282044836093, 4.56481798372561365305760113005, 4.80359661336207276368356103687, 5.32004334725503082438528115541, 5.38544076306636054804861513856, 5.42049599438066676918451385701, 5.70911749078672148659568727425, 6.01614542834553048334039879602, 6.11355043638659871035896916060, 6.30498621919867198686755985530, 6.33310456187507958406643792996, 6.95027774940935982446546029834, 6.96447000545605885432459726530

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