Properties

Label 2-76664-1.1-c1-0-5
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 − 2·5-s − 7-s + 9-s − 5·11-s − 6·13-s − 4·15-s + 2·17-s + 4·19-s − 2·21-s + 3·23-s − 25-s − 4·27-s − 7·29-s − 10·31-s − 10·33-s + 2·35-s − 12·39-s + 10·41-s − 4·43-s − 2·45-s + 6·47-s + 49-s + 4·51-s + 3·53-s + 10·55-s + 8·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 1.66·13-s − 1.03·15-s + 0.485·17-s + 0.917·19-s − 0.436·21-s + 0.625·23-s − 1/5·25-s − 0.769·27-s − 1.29·29-s − 1.79·31-s − 1.74·33-s + 0.338·35-s − 1.92·39-s + 1.56·41-s − 0.609·43-s − 0.298·45-s + 0.875·47-s + 1/7·49-s + 0.560·51-s + 0.412·53-s + 1.34·55-s + 1.05·57-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.38083855723058, −13.89689972590001, −13.17996081787927, −12.86399022836962, −12.49486297977436, −11.82965844154270, −11.25611094537815, −10.88440690351241, −10.03324599160310, −9.705134759192061, −9.310312146004431, −8.649236834033919, −8.082692805580262, −7.601053135754816, −7.340184933551333, −7.028497776321986, −5.699611753531111, −5.443892664554285, −4.907365691324356, −3.988961062801908, −3.581541425279492, −3.054995418778442, −2.341402334205129, −2.137326639143865, −0.7204266301550127, 0, 0.7204266301550127, 2.137326639143865, 2.341402334205129, 3.054995418778442, 3.581541425279492, 3.988961062801908, 4.907365691324356, 5.443892664554285, 5.699611753531111, 7.028497776321986, 7.340184933551333, 7.601053135754816, 8.082692805580262, 8.649236834033919, 9.310312146004431, 9.705134759192061, 10.03324599160310, 10.88440690351241, 11.25611094537815, 11.82965844154270, 12.49486297977436, 12.86399022836962, 13.17996081787927, 13.89689972590001, 14.38083855723058

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