Properties

Label 2-7800-1.1-c1-0-4
Degree $2$
Conductor $7800$
Sign $1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 4.64·11-s − 13-s − 4.24·17-s − 6.24·19-s − 2.24·23-s − 27-s − 9.21·29-s + 9.28·31-s + 4.64·33-s − 7.28·37-s + 39-s − 5.67·41-s + 4.24·43-s − 2.88·47-s − 7·49-s + 4.24·51-s + 9.21·53-s + 6.24·57-s + 5.92·59-s + 0.969·61-s + 1.93·67-s + 2.24·69-s + 5.60·71-s + 12.5·73-s + 12.2·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.333·9-s − 1.39·11-s − 0.277·13-s − 1.03·17-s − 1.43·19-s − 0.469·23-s − 0.192·27-s − 1.71·29-s + 1.66·31-s + 0.807·33-s − 1.19·37-s + 0.160·39-s − 0.885·41-s + 0.648·43-s − 0.421·47-s − 49-s + 0.595·51-s + 1.26·53-s + 0.827·57-s + 0.770·59-s + 0.124·61-s + 0.236·67-s + 0.270·69-s + 0.665·71-s + 1.47·73-s + 1.37·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(62.2833\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6190038081\)
\(L(\frac12)\) \(\approx\) \(0.6190038081\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 + 6.24T + 19T^{2} \)
23 \( 1 + 2.24T + 23T^{2} \)
29 \( 1 + 9.21T + 29T^{2} \)
31 \( 1 - 9.28T + 31T^{2} \)
37 \( 1 + 7.28T + 37T^{2} \)
41 \( 1 + 5.67T + 41T^{2} \)
43 \( 1 - 4.24T + 43T^{2} \)
47 \( 1 + 2.88T + 47T^{2} \)
53 \( 1 - 9.21T + 53T^{2} \)
59 \( 1 - 5.92T + 59T^{2} \)
61 \( 1 - 0.969T + 61T^{2} \)
67 \( 1 - 1.93T + 67T^{2} \)
71 \( 1 - 5.60T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 - 3.67T + 83T^{2} \)
89 \( 1 + 9.67T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−8.006969271006219103153679067393, −6.96538411088603795364213743129, −6.57616106872550525270366432391, −5.68698586867143787130507874959, −5.10106057390892458554905689707, −4.43099735751405480050994468139, −3.63362864948348296696776105558, −2.47150415270460808491945962002, −1.93893175193214366695245804105, −0.37878694050747900260301198063, 0.37878694050747900260301198063, 1.93893175193214366695245804105, 2.47150415270460808491945962002, 3.63362864948348296696776105558, 4.43099735751405480050994468139, 5.10106057390892458554905689707, 5.68698586867143787130507874959, 6.57616106872550525270366432391, 6.96538411088603795364213743129, 8.006969271006219103153679067393

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