Properties

Degree $10$
Conductor $1.330\times 10^{14}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 10·5-s + 9·7-s + 5·11-s − 4·13-s + 30·15-s − 2·17-s + 5·19-s + 27·21-s + 6·23-s + 35·25-s − 8·27-s − 5·29-s + 9·31-s + 15·33-s + 90·35-s + 8·37-s − 12·39-s − 4·41-s + 8·43-s + 13·47-s + 30·49-s − 6·51-s − 2·53-s + 50·55-s + 15·57-s + 4·59-s + ⋯
L(s)  = 1  + 1.73·3-s + 4.47·5-s + 3.40·7-s + 1.50·11-s − 1.10·13-s + 7.74·15-s − 0.485·17-s + 1.14·19-s + 5.89·21-s + 1.25·23-s + 7·25-s − 1.53·27-s − 0.928·29-s + 1.61·31-s + 2.61·33-s + 15.2·35-s + 1.31·37-s − 1.92·39-s − 0.624·41-s + 1.21·43-s + 1.89·47-s + 30/7·49-s − 0.840·51-s − 0.274·53-s + 6.74·55-s + 1.98·57-s + 0.520·59-s + ⋯

Functional equation

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

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 167^{5}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{668} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{10} \cdot 167^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(34.9557\)
\(L(\frac12)\) \(\approx\) \(34.9557\)
\(L(\frac{3}{2})\) 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 \( 1 \)
167$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - p T + p^{2} T^{2} - 19 T^{3} + 5 p^{2} T^{4} - 83 T^{5} + 5 p^{3} T^{6} - 19 p^{2} T^{7} + p^{5} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{5} \)
7$C_2 \wr S_5$ \( 1 - 9 T + 51 T^{2} - 223 T^{3} + 787 T^{4} - 2265 T^{5} + 787 p T^{6} - 223 p^{2} T^{7} + 51 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 5 T + 43 T^{2} - 191 T^{3} + 849 T^{4} - 3017 T^{5} + 849 p T^{6} - 191 p^{2} T^{7} + 43 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 4 T + 45 T^{2} + 160 T^{3} + 1006 T^{4} + 2840 T^{5} + 1006 p T^{6} + 160 p^{2} T^{7} + 45 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 2 T + 25 T^{2} + 96 T^{3} + 662 T^{4} + 1308 T^{5} + 662 p T^{6} + 96 p^{2} T^{7} + 25 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 5 T + 53 T^{2} - 279 T^{3} + 1821 T^{4} - 6547 T^{5} + 1821 p T^{6} - 279 p^{2} T^{7} + 53 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 6 T + 63 T^{2} - 192 T^{3} + 1414 T^{4} - 2452 T^{5} + 1414 p T^{6} - 192 p^{2} T^{7} + 63 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 5 T + 117 T^{2} + 541 T^{3} + 6005 T^{4} + 22981 T^{5} + 6005 p T^{6} + 541 p^{2} T^{7} + 117 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 9 T + 119 T^{2} - 909 T^{3} + 6257 T^{4} - 39425 T^{5} + 6257 p T^{6} - 909 p^{2} T^{7} + 119 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 8 T + 141 T^{2} - 1104 T^{3} + 8998 T^{4} - 60080 T^{5} + 8998 p T^{6} - 1104 p^{2} T^{7} + 141 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 4 T + 109 T^{2} + 424 T^{3} + 6474 T^{4} + 20392 T^{5} + 6474 p T^{6} + 424 p^{2} T^{7} + 109 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 8 T + 135 T^{2} - 992 T^{3} + 9706 T^{4} - 56752 T^{5} + 9706 p T^{6} - 992 p^{2} T^{7} + 135 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 13 T + 261 T^{2} - 2283 T^{3} + 539 p T^{4} - 157087 T^{5} + 539 p^{2} T^{6} - 2283 p^{2} T^{7} + 261 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 2 T + 141 T^{2} + 216 T^{3} + 11910 T^{4} + 17068 T^{5} + 11910 p T^{6} + 216 p^{2} T^{7} + 141 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 4 T + p T^{2} - 120 T^{3} + 4414 T^{4} - 36008 T^{5} + 4414 p T^{6} - 120 p^{2} T^{7} + p^{4} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 11 T + 229 T^{2} - 2491 T^{3} + 24489 T^{4} - 221163 T^{5} + 24489 p T^{6} - 2491 p^{2} T^{7} + 229 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 28 T + 491 T^{2} - 5616 T^{3} + 53302 T^{4} - 436232 T^{5} + 53302 p T^{6} - 5616 p^{2} T^{7} + 491 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 2 T + 175 T^{2} - 320 T^{3} + 17542 T^{4} - 36060 T^{5} + 17542 p T^{6} - 320 p^{2} T^{7} + 175 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 8 T + 241 T^{2} - 616 T^{3} + 19206 T^{4} + 4320 T^{5} + 19206 p T^{6} - 616 p^{2} T^{7} + 241 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 10 T + 139 T^{2} + 1552 T^{3} + 15962 T^{4} + 195500 T^{5} + 15962 p T^{6} + 1552 p^{2} T^{7} + 139 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 2 T + 179 T^{2} - 656 T^{3} + 20622 T^{4} - 69980 T^{5} + 20622 p T^{6} - 656 p^{2} T^{7} + 179 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 17 T + 485 T^{2} + 5689 T^{3} + 89097 T^{4} + 743149 T^{5} + 89097 p T^{6} + 5689 p^{2} T^{7} + 485 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 27 T + 521 T^{2} + 8059 T^{3} + 104425 T^{4} + 1093259 T^{5} + 104425 p T^{6} + 8059 p^{2} T^{7} + 521 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.31133820922531379592352314443, −6.07783765845692870716981724547, −6.01595222655079222294446645521, −5.72192644241367696862147939740, −5.69832933100411819944606081136, −5.34030650694394925755197301763, −5.19011943931668568990609980139, −5.15780096951604515523804045145, −5.10908223471499265317748868139, −4.74384023820616629854639779279, −4.31478346351207238068220125386, −4.28328384991421060527371329937, −3.87689014113227537308151753362, −3.76603271965913903382684735729, −3.58005696895977353115262993220, −2.89162945064397829985236151457, −2.56223257888144330032574909925, −2.53370254557324170740912158352, −2.47267815614508293982749650367, −2.42838613076373027708445734664, −1.87517241733764295673242023254, −1.79058045829787187667008636688, −1.45041119569858542143775263677, −1.22144523336725543105635312858, −1.12079404176756742640723923648, 1.12079404176756742640723923648, 1.22144523336725543105635312858, 1.45041119569858542143775263677, 1.79058045829787187667008636688, 1.87517241733764295673242023254, 2.42838613076373027708445734664, 2.47267815614508293982749650367, 2.53370254557324170740912158352, 2.56223257888144330032574909925, 2.89162945064397829985236151457, 3.58005696895977353115262993220, 3.76603271965913903382684735729, 3.87689014113227537308151753362, 4.28328384991421060527371329937, 4.31478346351207238068220125386, 4.74384023820616629854639779279, 5.10908223471499265317748868139, 5.15780096951604515523804045145, 5.19011943931668568990609980139, 5.34030650694394925755197301763, 5.69832933100411819944606081136, 5.72192644241367696862147939740, 6.01595222655079222294446645521, 6.07783765845692870716981724547, 6.31133820922531379592352314443

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