Properties

Label 2-7800-1.1-c1-0-35
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 + 4.83·7-s + 9-s + 2.25·11-s + 13-s − 6.32·17-s − 2.58·19-s − 4.83·21-s − 4.83·23-s − 27-s + 9.09·29-s − 0.510·31-s − 2.25·33-s − 4.25·37-s − 39-s + 0.255·41-s − 9.16·43-s + 4.51·47-s + 16.4·49-s + 6.32·51-s + 9.93·53-s + 2.58·57-s + 14.1·59-s + 9.93·61-s + 4.83·63-s + 13.1·67-s + 4.83·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.82·7-s + 0.333·9-s + 0.679·11-s + 0.277·13-s − 1.53·17-s − 0.592·19-s − 1.05·21-s − 1.00·23-s − 0.192·27-s + 1.68·29-s − 0.0916·31-s − 0.392·33-s − 0.699·37-s − 0.160·39-s + 0.0398·41-s − 1.39·43-s + 0.657·47-s + 2.34·49-s + 0.886·51-s + 1.36·53-s + 0.342·57-s + 1.84·59-s + 1.27·61-s + 0.609·63-s + 1.60·67-s + 0.582·69-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

\(L(1)\) \(\approx\) \(2.126550159\)
\(L(\frac12)\) \(\approx\) \(2.126550159\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.83T + 7T^{2} \)
11 \( 1 - 2.25T + 11T^{2} \)
17 \( 1 + 6.32T + 17T^{2} \)
19 \( 1 + 2.58T + 19T^{2} \)
23 \( 1 + 4.83T + 23T^{2} \)
29 \( 1 - 9.09T + 29T^{2} \)
31 \( 1 + 0.510T + 31T^{2} \)
37 \( 1 + 4.25T + 37T^{2} \)
41 \( 1 - 0.255T + 41T^{2} \)
43 \( 1 + 9.16T + 43T^{2} \)
47 \( 1 - 4.51T + 47T^{2} \)
53 \( 1 - 9.93T + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 - 9.93T + 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 + 5.74T + 71T^{2} \)
73 \( 1 + 2.90T + 73T^{2} \)
79 \( 1 - 5.74T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−8.011624286498535626592586537675, −6.93041136517787506798242368227, −6.63871240153221342085490185068, −5.66678247383174095845090383777, −5.01919880369671783130541656517, −4.34240483211921496672096916757, −3.89840924713107628819073531767, −2.35206918202470307670813960654, −1.77866166453167689708972984679, −0.77465680041494378385884869220, 0.77465680041494378385884869220, 1.77866166453167689708972984679, 2.35206918202470307670813960654, 3.89840924713107628819073531767, 4.34240483211921496672096916757, 5.01919880369671783130541656517, 5.66678247383174095845090383777, 6.63871240153221342085490185068, 6.93041136517787506798242368227, 8.011624286498535626592586537675

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