Properties

Label 2-7800-1.1-c1-0-110
Degree $2$
Conductor $7800$
Sign $-1$
Analytic cond. $62.2833$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 4·11-s − 13-s + 2·17-s − 4·19-s − 4·23-s + 27-s − 6·29-s − 4·31-s + 4·33-s − 6·37-s − 39-s − 2·41-s − 12·43-s − 8·47-s − 7·49-s + 2·51-s − 14·53-s − 4·57-s + 12·59-s − 2·61-s − 4·69-s + 8·71-s + 2·73-s + 8·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s − 0.986·37-s − 0.160·39-s − 0.312·41-s − 1.82·43-s − 1.16·47-s − 49-s + 0.280·51-s − 1.92·53-s − 0.529·57-s + 1.56·59-s − 0.256·61-s − 0.481·69-s + 0.949·71-s + 0.234·73-s + 0.900·79-s + 1/9·81-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: yes
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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.59499942265741132817665548039, −6.73262641164950051142451665078, −6.35761349768140561273772871158, −5.34770733733994876965528529787, −4.61611270593743150732647425788, −3.68722841172202400700992394977, −3.36892009313603717025518168272, −2.04891969776081833544716378077, −1.56123427242982466090595067309, 0, 1.56123427242982466090595067309, 2.04891969776081833544716378077, 3.36892009313603717025518168272, 3.68722841172202400700992394977, 4.61611270593743150732647425788, 5.34770733733994876965528529787, 6.35761349768140561273772871158, 6.73262641164950051142451665078, 7.59499942265741132817665548039

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