Properties

Label 2-64400-1.1-c1-0-25
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s − 2·9-s + 2·11-s + 5·13-s + 6·17-s + 6·19-s + 21-s + 23-s + 5·27-s − 3·29-s + 3·31-s − 2·33-s − 5·39-s − 7·41-s − 6·43-s − 9·47-s + 49-s − 6·51-s − 6·57-s + 4·59-s − 4·61-s + 2·63-s + 8·67-s − 69-s + 13·71-s + 11·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.603·11-s + 1.38·13-s + 1.45·17-s + 1.37·19-s + 0.218·21-s + 0.208·23-s + 0.962·27-s − 0.557·29-s + 0.538·31-s − 0.348·33-s − 0.800·39-s − 1.09·41-s − 0.914·43-s − 1.31·47-s + 1/7·49-s − 0.840·51-s − 0.794·57-s + 0.520·59-s − 0.512·61-s + 0.251·63-s + 0.977·67-s − 0.120·69-s + 1.54·71-s + 1.28·73-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: $\chi_{64400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.315473673\)
\(L(\frac12)\) \(\approx\) \(2.315473673\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.16011414211293, −13.72620816916916, −13.32533684512470, −12.64952158284119, −12.00317222640647, −11.80700750496671, −11.22669880019288, −10.85829550483233, −10.09248987097574, −9.707898106995557, −9.206792020685851, −8.454587363377591, −8.171868269540852, −7.490208657538081, −6.713843370466336, −6.395053749364739, −5.801276907265562, −5.277740079276403, −4.897746901064124, −3.790353186529263, −3.406822724439242, −3.048297218727345, −1.907610779052632, −1.139754366706531, −0.6230785004303801, 0.6230785004303801, 1.139754366706531, 1.907610779052632, 3.048297218727345, 3.406822724439242, 3.790353186529263, 4.897746901064124, 5.277740079276403, 5.801276907265562, 6.395053749364739, 6.713843370466336, 7.490208657538081, 8.171868269540852, 8.454587363377591, 9.206792020685851, 9.707898106995557, 10.09248987097574, 10.85829550483233, 11.22669880019288, 11.80700750496671, 12.00317222640647, 12.64952158284119, 13.32533684512470, 13.72620816916916, 14.16011414211293

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