Properties

Degree $2$
Conductor $350658$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 2·13-s − 14-s + 16-s − 6·17-s − 4·19-s + 2·20-s + 23-s − 25-s + 2·26-s − 28-s − 2·29-s − 8·31-s + 32-s − 6·34-s − 2·35-s + 6·37-s − 4·38-s + 2·40-s − 6·41-s + 4·43-s + 46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.447·20-s + 0.208·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.338·35-s + 0.986·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s + 0.609·43-s + 0.147·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350658 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350658 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350658\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11^{2} \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{350658} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 350658,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
23 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.77080452383078, −12.64756241526426, −11.89623169046532, −11.34054891339026, −10.95844297006785, −10.70094200105185, −10.11941166975301, −9.479499116490601, −9.292573750016852, −8.671455432570419, −8.285291011994728, −7.591127135245977, −7.038183970948026, −6.615469775443001, −6.253921911946741, −5.707740587618762, −5.453861012173577, −4.690414269912738, −4.269530678811020, −3.766621969388987, −3.261803258412725, −2.485485508456682, −2.118984620494349, −1.727378424972155, −0.8303759589739566, 0, 0.8303759589739566, 1.727378424972155, 2.118984620494349, 2.485485508456682, 3.261803258412725, 3.766621969388987, 4.269530678811020, 4.690414269912738, 5.453861012173577, 5.707740587618762, 6.253921911946741, 6.615469775443001, 7.038183970948026, 7.591127135245977, 8.285291011994728, 8.671455432570419, 9.292573750016852, 9.479499116490601, 10.11941166975301, 10.70094200105185, 10.95844297006785, 11.34054891339026, 11.89623169046532, 12.64756241526426, 12.77080452383078

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