Properties

Label 2-76664-1.1-c1-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s − 7-s + 9-s + 4·11-s + 6·13-s + 4·15-s + 4·17-s − 4·19-s − 2·21-s − 25-s − 4·27-s + 10·29-s − 2·31-s + 8·33-s − 2·35-s + 12·39-s − 2·41-s + 4·43-s + 2·45-s + 49-s + 8·51-s + 6·53-s + 8·55-s − 8·57-s + 14·61-s − 63-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 1.03·15-s + 0.970·17-s − 0.917·19-s − 0.436·21-s − 1/5·25-s − 0.769·27-s + 1.85·29-s − 0.359·31-s + 1.39·33-s − 0.338·35-s + 1.92·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 1.12·51-s + 0.824·53-s + 1.07·55-s − 1.05·57-s + 1.79·61-s − 0.125·63-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: \(0\)
Selberg data: \((2,\ 76664,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.353877434\)
\(L(\frac12)\) \(\approx\) \(6.353877434\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.11628878401155, −13.53794206251459, −13.28050610817417, −12.75880903543865, −12.04594558175807, −11.62800648615292, −10.98280188707268, −10.32845434693108, −9.945001989093984, −9.456473148649747, −8.841164422213017, −8.567727153989461, −8.230563374529070, −7.381708687704985, −6.777682248944444, −6.201197743639283, −5.933875683741462, −5.270472982900398, −4.221333909523781, −3.904052484611629, −3.309427200264676, −2.746296081571402, −2.040435205645041, −1.438118329627934, −0.8052891854472825, 0.8052891854472825, 1.438118329627934, 2.040435205645041, 2.746296081571402, 3.309427200264676, 3.904052484611629, 4.221333909523781, 5.270472982900398, 5.933875683741462, 6.201197743639283, 6.777682248944444, 7.381708687704985, 8.230563374529070, 8.567727153989461, 8.841164422213017, 9.456473148649747, 9.945001989093984, 10.32845434693108, 10.98280188707268, 11.62800648615292, 12.04594558175807, 12.75880903543865, 13.28050610817417, 13.53794206251459, 14.11628878401155

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