Properties

Label 2-7800-1.1-c1-0-112
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 + 2.82·7-s + 9-s + 1.05·11-s − 13-s − 7.14·17-s − 6.61·19-s + 2.82·21-s + 3.49·23-s + 27-s + 4.82·29-s − 9.65·31-s + 1.05·33-s − 8.11·37-s − 39-s − 3.26·41-s + 7.44·43-s − 11.3·47-s + 1.00·49-s − 7.14·51-s − 4.53·53-s − 6.61·57-s + 1.05·59-s − 7.97·61-s + 2.82·63-s − 1.46·67-s + 3.49·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.06·7-s + 0.333·9-s + 0.318·11-s − 0.277·13-s − 1.73·17-s − 1.51·19-s + 0.617·21-s + 0.728·23-s + 0.192·27-s + 0.896·29-s − 1.73·31-s + 0.183·33-s − 1.33·37-s − 0.160·39-s − 0.510·41-s + 1.13·43-s − 1.65·47-s + 0.142·49-s − 1.00·51-s − 0.622·53-s − 0.876·57-s + 0.137·59-s − 1.02·61-s + 0.356·63-s − 0.179·67-s + 0.420·69-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: \(1\)
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 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 - 1.05T + 11T^{2} \)
17 \( 1 + 7.14T + 17T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 - 3.49T + 23T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + 9.65T + 31T^{2} \)
37 \( 1 + 8.11T + 37T^{2} \)
41 \( 1 + 3.26T + 41T^{2} \)
43 \( 1 - 7.44T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + 4.53T + 53T^{2} \)
59 \( 1 - 1.05T + 59T^{2} \)
61 \( 1 + 7.97T + 61T^{2} \)
67 \( 1 + 1.46T + 67T^{2} \)
71 \( 1 - 0.845T + 71T^{2} \)
73 \( 1 + 9.28T + 73T^{2} \)
79 \( 1 - 6.85T + 79T^{2} \)
83 \( 1 + 7.97T + 83T^{2} \)
89 \( 1 - 0.139T + 89T^{2} \)
97 \( 1 - 6.93T + 97T^{2} \)
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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

−7.55369088412552084722585787808, −6.84590852602777166863171811365, −6.32807964807604178054569986568, −5.19509844778754064970109531447, −4.60346793497560423384956087771, −4.05726985802655190485671254093, −3.04659524702784884811691368653, −2.08187610881511696666211467340, −1.60853851117380814363808442063, 0, 1.60853851117380814363808442063, 2.08187610881511696666211467340, 3.04659524702784884811691368653, 4.05726985802655190485671254093, 4.60346793497560423384956087771, 5.19509844778754064970109531447, 6.32807964807604178054569986568, 6.84590852602777166863171811365, 7.55369088412552084722585787808

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