Properties

Label 2-76664-1.1-c1-0-0
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  − 7-s − 3·9-s + 4·11-s + 4·17-s − 2·19-s + 4·23-s − 5·25-s − 10·29-s + 6·31-s − 6·41-s − 4·43-s − 12·47-s + 49-s − 6·53-s + 2·59-s − 12·61-s + 3·63-s − 12·67-s − 8·71-s + 2·73-s − 4·77-s + 12·79-s + 9·81-s − 16·83-s − 12·89-s + 8·97-s − 12·99-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s + 1.20·11-s + 0.970·17-s − 0.458·19-s + 0.834·23-s − 25-s − 1.85·29-s + 1.07·31-s − 0.937·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.260·59-s − 1.53·61-s + 0.377·63-s − 1.46·67-s − 0.949·71-s + 0.234·73-s − 0.455·77-s + 1.35·79-s + 81-s − 1.75·83-s − 1.27·89-s + 0.812·97-s − 1.20·99-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

\(L(1)\) \(\approx\) \(1.146755953\)
\(L(\frac12)\) \(\approx\) \(1.146755953\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−14.06075306122454, −13.47173445407280, −13.23126957435153, −12.37050038140323, −12.10076604083860, −11.42556577682403, −11.29671856960058, −10.55556822845576, −9.840007042968128, −9.573707494927366, −8.995061095191942, −8.486200491201077, −7.991709994803612, −7.355278054366115, −6.806285911772331, −6.094668343847271, −5.950389988252128, −5.173807873974492, −4.574728769659888, −3.847034825140431, −3.263010394734153, −2.953265068011629, −1.857750890687500, −1.433540710048625, −0.3459746332200766, 0.3459746332200766, 1.433540710048625, 1.857750890687500, 2.953265068011629, 3.263010394734153, 3.847034825140431, 4.574728769659888, 5.173807873974492, 5.950389988252128, 6.094668343847271, 6.806285911772331, 7.355278054366115, 7.991709994803612, 8.486200491201077, 8.995061095191942, 9.573707494927366, 9.840007042968128, 10.55556822845576, 11.29671856960058, 11.42556577682403, 12.10076604083860, 12.37050038140323, 13.23126957435153, 13.47173445407280, 14.06075306122454

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