Properties

Label 2-7800-1.1-c1-0-71
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 − 3.09·7-s + 9-s + 5.51·11-s + 13-s − 6.21·17-s + 3.09·21-s + 4.21·23-s − 27-s − 1.70·29-s − 5.51·33-s − 4.02·37-s − 39-s + 0.198·41-s − 5.70·43-s − 6.41·47-s + 2.57·49-s + 6.21·51-s + 4.02·53-s − 7.53·59-s + 11.9·61-s − 3.09·63-s + 4.83·67-s − 4.21·69-s + 7.98·71-s + 9.31·73-s − 17.0·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.16·7-s + 0.333·9-s + 1.66·11-s + 0.277·13-s − 1.50·17-s + 0.675·21-s + 0.879·23-s − 0.192·27-s − 0.317·29-s − 0.959·33-s − 0.662·37-s − 0.160·39-s + 0.0310·41-s − 0.870·43-s − 0.935·47-s + 0.367·49-s + 0.870·51-s + 0.553·53-s − 0.981·59-s + 1.52·61-s − 0.389·63-s + 0.590·67-s − 0.507·69-s + 0.948·71-s + 1.08·73-s − 1.94·77-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 \( 1 + 3.09T + 7T^{2} \)
11 \( 1 - 5.51T + 11T^{2} \)
17 \( 1 + 6.21T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4.21T + 23T^{2} \)
29 \( 1 + 1.70T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 4.02T + 37T^{2} \)
41 \( 1 - 0.198T + 41T^{2} \)
43 \( 1 + 5.70T + 43T^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 - 4.02T + 53T^{2} \)
59 \( 1 + 7.53T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 4.83T + 67T^{2} \)
71 \( 1 - 7.98T + 71T^{2} \)
73 \( 1 - 9.31T + 73T^{2} \)
79 \( 1 - 8.51T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + 9.40T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−7.13461518397629496767852599662, −6.59180060547610904152808747057, −6.46985791764281534132065196853, −5.49017157152100285261890666626, −4.65001056103789450696785205152, −3.86972592616098590059244680699, −3.32074589151223743370057936121, −2.16090022726238213684115631420, −1.14164525413407941619893673049, 0, 1.14164525413407941619893673049, 2.16090022726238213684115631420, 3.32074589151223743370057936121, 3.86972592616098590059244680699, 4.65001056103789450696785205152, 5.49017157152100285261890666626, 6.46985791764281534132065196853, 6.59180060547610904152808747057, 7.13461518397629496767852599662

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