Properties

Label 2-7800-1.1-c1-0-30
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 − 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: \(0\)
Selberg data: \((2,\ 7800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.115392379\)
\(L(\frac12)\) \(\approx\) \(2.115392379\)
\(L(\frac{3}{2})\) 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

−7.898601872183510745471942063650, −7.07887586582490824219196118760, −6.37412077137627625529700486489, −6.15005817220053809787088867415, −4.95515973118666087053754639419, −4.03998480593778221413283022259, −3.54836061815997226057730920375, −2.79790544488079651684554345416, −1.87353633001064472785425594687, −0.69762297965155179851538211371, 0.69762297965155179851538211371, 1.87353633001064472785425594687, 2.79790544488079651684554345416, 3.54836061815997226057730920375, 4.03998480593778221413283022259, 4.95515973118666087053754639419, 6.15005817220053809787088867415, 6.37412077137627625529700486489, 7.07887586582490824219196118760, 7.898601872183510745471942063650

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