Properties

Label 2-64400-1.1-c1-0-12
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 − 5·11-s − 13-s + 3·17-s + 4·19-s + 21-s − 23-s + 5·27-s − 29-s + 10·31-s + 5·33-s + 4·37-s + 39-s + 12·41-s − 6·43-s + 11·47-s + 49-s − 3·51-s + 14·53-s − 4·57-s − 10·59-s − 6·61-s + 2·63-s + 8·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.50·11-s − 0.277·13-s + 0.727·17-s + 0.917·19-s + 0.218·21-s − 0.208·23-s + 0.962·27-s − 0.185·29-s + 1.79·31-s + 0.870·33-s + 0.657·37-s + 0.160·39-s + 1.87·41-s − 0.914·43-s + 1.60·47-s + 1/7·49-s − 0.420·51-s + 1.92·53-s − 0.529·57-s − 1.30·59-s − 0.768·61-s + 0.251·63-s + 0.977·67-s + 0.120·69-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

\(L(1)\) \(\approx\) \(1.512809321\)
\(L(\frac12)\) \(\approx\) \(1.512809321\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.06729817740480, −13.71547114545856, −13.33544796328630, −12.48979817601243, −12.27889486677291, −11.81225356262635, −11.10624719305403, −10.75198829826100, −10.20591401699924, −9.728267632067407, −9.246387358117914, −8.459365409659640, −7.951158866904140, −7.602145434309998, −6.929785557150982, −6.223264433622966, −5.735487621843888, −5.341551701157455, −4.815018554773784, −4.094238035174399, −3.263772778715518, −2.685399255182306, −2.344364050636486, −0.9973737188340008, −0.5225942634168226, 0.5225942634168226, 0.9973737188340008, 2.344364050636486, 2.685399255182306, 3.263772778715518, 4.094238035174399, 4.815018554773784, 5.341551701157455, 5.735487621843888, 6.223264433622966, 6.929785557150982, 7.602145434309998, 7.951158866904140, 8.459365409659640, 9.246387358117914, 9.728267632067407, 10.20591401699924, 10.75198829826100, 11.10624719305403, 11.81225356262635, 12.27889486677291, 12.48979817601243, 13.33544796328630, 13.71547114545856, 14.06729817740480

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