Properties

Degree $2$
Conductor $41650$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s + 6·11-s − 2·12-s + 2·13-s + 16-s − 17-s − 18-s + 4·19-s − 6·22-s + 2·24-s − 2·26-s + 4·27-s + 4·31-s − 32-s − 12·33-s + 34-s + 36-s + 4·37-s − 4·38-s − 4·39-s − 6·41-s − 8·43-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 1.27·22-s + 0.408·24-s − 0.392·26-s + 0.769·27-s + 0.718·31-s − 0.176·32-s − 2.08·33-s + 0.171·34-s + 1/6·36-s + 0.657·37-s − 0.648·38-s − 0.640·39-s − 0.937·41-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41650\)    =    \(2 \cdot 5^{2} \cdot 7^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{41650} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 41650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.260412385\)
\(L(\frac12)\) \(\approx\) \(1.260412385\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
17 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.96179195459949, −14.21675696426482, −13.70691690525711, −13.17981817260769, −12.27846982909289, −11.91307463325978, −11.63888009746653, −11.16771703111331, −10.61158617082302, −10.00435190330315, −9.512975101667060, −8.941640436446410, −8.462880597741316, −7.835254991926610, −6.937645637791402, −6.689955952949159, −6.183087765425497, −5.617332847908276, −4.965745352632205, −4.243097566632904, −3.576629723505404, −2.883866626474754, −1.795324388763474, −1.170734226088669, −0.5741187747808603, 0.5741187747808603, 1.170734226088669, 1.795324388763474, 2.883866626474754, 3.576629723505404, 4.243097566632904, 4.965745352632205, 5.617332847908276, 6.183087765425497, 6.689955952949159, 6.937645637791402, 7.835254991926610, 8.462880597741316, 8.941640436446410, 9.512975101667060, 10.00435190330315, 10.61158617082302, 11.16771703111331, 11.63888009746653, 11.91307463325978, 12.27846982909289, 13.17981817260769, 13.70691690525711, 14.21675696426482, 14.96179195459949

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