Properties

Label 2-76664-1.1-c1-0-10
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  + 2·3-s + 4·5-s + 7-s + 9-s + 8·15-s + 2·17-s + 2·19-s + 2·21-s − 8·23-s + 11·25-s − 4·27-s − 2·29-s − 4·31-s + 4·35-s − 2·41-s − 8·43-s + 4·45-s − 4·47-s + 49-s + 4·51-s − 10·53-s + 4·57-s − 6·59-s − 4·61-s + 63-s − 12·67-s − 16·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 2.06·15-s + 0.485·17-s + 0.458·19-s + 0.436·21-s − 1.66·23-s + 11/5·25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s + 0.676·35-s − 0.312·41-s − 1.21·43-s + 0.596·45-s − 0.583·47-s + 1/7·49-s + 0.560·51-s − 1.37·53-s + 0.529·57-s − 0.781·59-s − 0.512·61-s + 0.125·63-s − 1.46·67-s − 1.92·69-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 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.35801527375651, −13.73691923742483, −13.51472504377908, −13.07969604125005, −12.40725035058110, −11.87638824146435, −11.26878919966318, −10.52462661375774, −10.17143743000642, −9.627143070018867, −9.309255732420704, −8.835430254452259, −8.211219474016051, −7.788978852493099, −7.232784510593116, −6.404562834456506, −6.013867500952746, −5.487400937327903, −4.953425211695578, −4.223938782151136, −3.388799988990764, −3.004312195301771, −2.277865305321467, −1.700273891404453, −1.464899323184333, 0, 1.464899323184333, 1.700273891404453, 2.277865305321467, 3.004312195301771, 3.388799988990764, 4.223938782151136, 4.953425211695578, 5.487400937327903, 6.013867500952746, 6.404562834456506, 7.232784510593116, 7.788978852493099, 8.211219474016051, 8.835430254452259, 9.309255732420704, 9.627143070018867, 10.17143743000642, 10.52462661375774, 11.26878919966318, 11.87638824146435, 12.40725035058110, 13.07969604125005, 13.51472504377908, 13.73691923742483, 14.35801527375651

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