Properties

Label 2-64400-1.1-c1-0-59
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  + 3-s + 7-s − 2·9-s + 6·11-s + 13-s − 6·17-s − 2·19-s + 21-s − 23-s − 5·27-s + 9·29-s + 31-s + 6·33-s − 8·37-s + 39-s + 9·41-s + 2·43-s − 3·47-s + 49-s − 6·51-s − 2·57-s − 12·59-s + 8·61-s − 2·63-s + 8·67-s − 69-s − 9·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.80·11-s + 0.277·13-s − 1.45·17-s − 0.458·19-s + 0.218·21-s − 0.208·23-s − 0.962·27-s + 1.67·29-s + 0.179·31-s + 1.04·33-s − 1.31·37-s + 0.160·39-s + 1.40·41-s + 0.304·43-s − 0.437·47-s + 1/7·49-s − 0.840·51-s − 0.264·57-s − 1.56·59-s + 1.02·61-s − 0.251·63-s + 0.977·67-s − 0.120·69-s − 1.06·71-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: \(1\)
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 \( 1 - T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.33472733643058, −14.23516977439694, −13.50811322204148, −13.20943084357263, −12.22308849366926, −12.09660713372026, −11.41789687295225, −11.01942884983418, −10.53575398293010, −9.758131590166261, −9.126100224403782, −8.934224885259926, −8.355006019148320, −8.036631996722214, −7.034065260810454, −6.738444954411967, −6.161660581629533, −5.653266579091241, −4.708049923941957, −4.281415973497743, −3.800742782148110, −3.029404874885442, −2.436511844557115, −1.738818323631002, −1.079872555942339, 0, 1.079872555942339, 1.738818323631002, 2.436511844557115, 3.029404874885442, 3.800742782148110, 4.281415973497743, 4.708049923941957, 5.653266579091241, 6.161660581629533, 6.738444954411967, 7.034065260810454, 8.036631996722214, 8.355006019148320, 8.934224885259926, 9.126100224403782, 9.758131590166261, 10.53575398293010, 11.01942884983418, 11.41789687295225, 12.09660713372026, 12.22308849366926, 13.20943084357263, 13.50811322204148, 14.23516977439694, 14.33472733643058

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