Properties

Label 2-7800-1.1-c1-0-65
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 − 4.88·7-s + 9-s + 6.11·11-s − 13-s + 2.88·17-s − 7.00·19-s + 4.88·21-s − 4.88·23-s − 27-s − 3.23·29-s + 9.77·31-s − 6.11·33-s + 9.89·37-s + 39-s − 9.89·41-s − 4·43-s + 5.77·47-s + 16.8·49-s − 2.88·51-s + 5.65·53-s + 7.00·57-s + 12.1·61-s − 4.88·63-s − 6.46·67-s + 4.88·69-s − 11.8·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.84·7-s + 0.333·9-s + 1.84·11-s − 0.277·13-s + 0.700·17-s − 1.60·19-s + 1.06·21-s − 1.01·23-s − 0.192·27-s − 0.599·29-s + 1.75·31-s − 1.06·33-s + 1.62·37-s + 0.160·39-s − 1.54·41-s − 0.609·43-s + 0.842·47-s + 2.41·49-s − 0.404·51-s + 0.777·53-s + 0.928·57-s + 1.55·61-s − 0.615·63-s − 0.789·67-s + 0.588·69-s − 1.41·71-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 \( 1 + 4.88T + 7T^{2} \)
11 \( 1 - 6.11T + 11T^{2} \)
17 \( 1 - 2.88T + 17T^{2} \)
19 \( 1 + 7.00T + 19T^{2} \)
23 \( 1 + 4.88T + 23T^{2} \)
29 \( 1 + 3.23T + 29T^{2} \)
31 \( 1 - 9.77T + 31T^{2} \)
37 \( 1 - 9.89T + 37T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 5.77T + 47T^{2} \)
53 \( 1 - 5.65T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 6.46T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 1.88T + 79T^{2} \)
83 \( 1 - 2.46T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + 5.11T + 97T^{2} \)
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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

−7.19754059397209531726990316454, −6.65431305468119600928870934547, −6.18168776703387905654728020822, −5.78395954036553390677787135410, −4.44173500626481641236939759559, −3.99389288392653420459943815428, −3.25168627037502626658719082789, −2.25332611827170463168943344081, −1.06413965304453787467947673615, 0, 1.06413965304453787467947673615, 2.25332611827170463168943344081, 3.25168627037502626658719082789, 3.99389288392653420459943815428, 4.44173500626481641236939759559, 5.78395954036553390677787135410, 6.18168776703387905654728020822, 6.65431305468119600928870934547, 7.19754059397209531726990316454

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