Properties

Label 2-7800-1.1-c1-0-49
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 + 2.76·7-s + 9-s + 4.49·11-s + 13-s + 3.62·17-s − 6.14·19-s − 2.76·21-s + 5.52·23-s − 27-s + 6.25·29-s + 8.49·31-s − 4.49·33-s − 0.896·37-s − 39-s + 2.89·41-s + 0.270·43-s + 3.38·47-s + 0.626·49-s − 3.62·51-s − 3.27·53-s + 6.14·57-s − 9.53·59-s + 11.4·61-s + 2.76·63-s − 12.4·67-s − 5.52·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.04·7-s + 0.333·9-s + 1.35·11-s + 0.277·13-s + 0.879·17-s − 1.41·19-s − 0.602·21-s + 1.15·23-s − 0.192·27-s + 1.16·29-s + 1.52·31-s − 0.781·33-s − 0.147·37-s − 0.160·39-s + 0.452·41-s + 0.0412·43-s + 0.494·47-s + 0.0894·49-s − 0.507·51-s − 0.449·53-s + 0.814·57-s − 1.24·59-s + 1.46·61-s + 0.347·63-s − 1.51·67-s − 0.664·69-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

\(L(1)\) \(\approx\) \(2.432650886\)
\(L(\frac12)\) \(\approx\) \(2.432650886\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.76T + 7T^{2} \)
11 \( 1 - 4.49T + 11T^{2} \)
17 \( 1 - 3.62T + 17T^{2} \)
19 \( 1 + 6.14T + 19T^{2} \)
23 \( 1 - 5.52T + 23T^{2} \)
29 \( 1 - 6.25T + 29T^{2} \)
31 \( 1 - 8.49T + 31T^{2} \)
37 \( 1 + 0.896T + 37T^{2} \)
41 \( 1 - 2.89T + 41T^{2} \)
43 \( 1 - 0.270T + 43T^{2} \)
47 \( 1 - 3.38T + 47T^{2} \)
53 \( 1 + 3.27T + 53T^{2} \)
59 \( 1 + 9.53T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 8.89T + 71T^{2} \)
73 \( 1 - 3.25T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 3.03T + 83T^{2} \)
89 \( 1 + 2.20T + 89T^{2} \)
97 \( 1 + 17.7T + 97T^{2} \)
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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

−7.983558325108707744759040585001, −6.97688020997106305419441151109, −6.48656465712215944311845932937, −5.85785052532985411325202301245, −4.89801923147723076273300320232, −4.48139705919317820498532377454, −3.69453926412329594451076877470, −2.62199306089391723577718638440, −1.50108316716807401137140963875, −0.907818607186173803066150140910, 0.907818607186173803066150140910, 1.50108316716807401137140963875, 2.62199306089391723577718638440, 3.69453926412329594451076877470, 4.48139705919317820498532377454, 4.89801923147723076273300320232, 5.85785052532985411325202301245, 6.48656465712215944311845932937, 6.97688020997106305419441151109, 7.983558325108707744759040585001

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