Properties

Label 2-64400-1.1-c1-0-33
Degree $2$
Conductor $64400$
Sign $-1$
Analytic cond. $514.236$
Root an. cond. $22.6767$
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  + 7-s − 3·9-s − 4·11-s − 3·13-s − 17-s + 23-s + 4·31-s + 11·37-s − 10·41-s + 2·43-s − 11·47-s + 49-s + 53-s + 8·59-s − 8·61-s − 3·63-s + 4·71-s − 4·73-s − 4·77-s + 11·79-s + 9·81-s + 13·83-s + 89-s − 3·91-s − 7·97-s + 12·99-s + 101-s + ⋯
L(s)  = 1  + 0.377·7-s − 9-s − 1.20·11-s − 0.832·13-s − 0.242·17-s + 0.208·23-s + 0.718·31-s + 1.80·37-s − 1.56·41-s + 0.304·43-s − 1.60·47-s + 1/7·49-s + 0.137·53-s + 1.04·59-s − 1.02·61-s − 0.377·63-s + 0.474·71-s − 0.468·73-s − 0.455·77-s + 1.23·79-s + 81-s + 1.42·83-s + 0.105·89-s − 0.314·91-s − 0.710·97-s + 1.20·99-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64400\)    =    \(2^{4} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(514.236\)
Root analytic conductor: \(22.6767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.60901069474202, −13.90871605053317, −13.51693781025779, −13.02691920303076, −12.49244131290553, −11.86161932189599, −11.48839908333899, −10.99249681001692, −10.44478964133731, −9.923782461360197, −9.442514217180024, −8.737053966521865, −8.261816321230843, −7.815443571298131, −7.398407806308400, −6.524192195060701, −6.171028152273942, −5.321167073805485, −5.034180964047580, −4.536197247899587, −3.629732723612799, −2.918510462122134, −2.509995773446112, −1.868088003092780, −0.7676129791313198, 0, 0.7676129791313198, 1.868088003092780, 2.509995773446112, 2.918510462122134, 3.629732723612799, 4.536197247899587, 5.034180964047580, 5.321167073805485, 6.171028152273942, 6.524192195060701, 7.398407806308400, 7.815443571298131, 8.261816321230843, 8.737053966521865, 9.442514217180024, 9.923782461360197, 10.44478964133731, 10.99249681001692, 11.48839908333899, 11.86161932189599, 12.49244131290553, 13.02691920303076, 13.51693781025779, 13.90871605053317, 14.60901069474202

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