Properties

Label 2-76664-1.1-c1-0-8
Degree $2$
Conductor $76664$
Sign $-1$
Analytic cond. $612.165$
Root an. cond. $24.7419$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 7-s − 3·9-s − 4·11-s + 4·13-s − 2·19-s + 11·25-s + 6·29-s − 2·31-s − 4·35-s + 2·41-s − 8·43-s − 12·45-s + 4·47-s + 49-s − 6·53-s − 16·55-s − 6·59-s + 3·63-s + 16·65-s − 4·67-s − 8·71-s + 10·73-s + 4·77-s + 8·79-s + 9·81-s − 16·83-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.377·7-s − 9-s − 1.20·11-s + 1.10·13-s − 0.458·19-s + 11/5·25-s + 1.11·29-s − 0.359·31-s − 0.676·35-s + 0.312·41-s − 1.21·43-s − 1.78·45-s + 0.583·47-s + 1/7·49-s − 0.824·53-s − 2.15·55-s − 0.781·59-s + 0.377·63-s + 1.98·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s + 0.900·79-s + 81-s − 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76664\)    =    \(2^{3} \cdot 7 \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(612.165\)
Root analytic conductor: \(24.7419\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{76664} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 76664,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
37 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.26928809904042, −13.63555462639819, −13.38675957338565, −13.01595480119721, −12.43298816893973, −11.85333953874005, −11.04074408765519, −10.72341371925128, −10.32990192100919, −9.803434381866981, −9.212547454364019, −8.814736409212699, −8.284054632930330, −7.809967640776665, −6.817772825718511, −6.474963252987528, −5.878333226539689, −5.629916820939435, −5.038046114891356, −4.423875203901331, −3.364854961816770, −2.950246616317754, −2.383965524788579, −1.787909184859684, −0.9976463254093583, 0, 0.9976463254093583, 1.787909184859684, 2.383965524788579, 2.950246616317754, 3.364854961816770, 4.423875203901331, 5.038046114891356, 5.629916820939435, 5.878333226539689, 6.474963252987528, 6.817772825718511, 7.809967640776665, 8.284054632930330, 8.814736409212699, 9.212547454364019, 9.803434381866981, 10.32990192100919, 10.72341371925128, 11.04074408765519, 11.85333953874005, 12.43298816893973, 13.01595480119721, 13.38675957338565, 13.63555462639819, 14.26928809904042

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