Properties

Label 2-7800-1.1-c1-0-101
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3.71·7-s + 9-s + 3.31·11-s − 13-s − 1.55·17-s − 5.33·19-s − 3.71·21-s − 0.442·23-s − 27-s + 2.56·29-s + 0.613·31-s − 3.31·33-s + 0.257·37-s + 39-s − 10.6·41-s − 12.6·43-s − 7.44·47-s + 6.78·49-s + 1.55·51-s + 5.39·53-s + 5.33·57-s − 13.1·59-s − 5.27·61-s + 3.71·63-s − 10.5·67-s + 0.442·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.40·7-s + 0.333·9-s + 1.00·11-s − 0.277·13-s − 0.377·17-s − 1.22·19-s − 0.810·21-s − 0.0923·23-s − 0.192·27-s + 0.477·29-s + 0.110·31-s − 0.577·33-s + 0.0423·37-s + 0.160·39-s − 1.65·41-s − 1.93·43-s − 1.08·47-s + 0.968·49-s + 0.218·51-s + 0.741·53-s + 0.706·57-s − 1.71·59-s − 0.675·61-s + 0.467·63-s − 1.28·67-s + 0.0533·69-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: \(1\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 3.71T + 7T^{2} \)
11 \( 1 - 3.31T + 11T^{2} \)
17 \( 1 + 1.55T + 17T^{2} \)
19 \( 1 + 5.33T + 19T^{2} \)
23 \( 1 + 0.442T + 23T^{2} \)
29 \( 1 - 2.56T + 29T^{2} \)
31 \( 1 - 0.613T + 31T^{2} \)
37 \( 1 - 0.257T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 + 7.44T + 47T^{2} \)
53 \( 1 - 5.39T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 + 5.27T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 - 0.311T + 71T^{2} \)
73 \( 1 + 9.46T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 4.58T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−7.46206282589971135574564211691, −6.69321687100261338100828072766, −6.25637727290144954301698026710, −5.28217183400205232555186448704, −4.65853824553594602729652136271, −4.22614889716342551702385045995, −3.15881780243801783823332121680, −1.88483738084071435534122147354, −1.44951328336009430052689816005, 0, 1.44951328336009430052689816005, 1.88483738084071435534122147354, 3.15881780243801783823332121680, 4.22614889716342551702385045995, 4.65853824553594602729652136271, 5.28217183400205232555186448704, 6.25637727290144954301698026710, 6.69321687100261338100828072766, 7.46206282589971135574564211691

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