Properties

Label 2-7800-1.1-c1-0-102
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 + 0.414·7-s + 9-s − 2.41·11-s + 13-s + 1.82·17-s − 2·19-s + 0.414·21-s − 4.82·23-s + 27-s + 2.65·29-s + 1.58·31-s − 2.41·33-s − 3.65·37-s + 39-s − 5.65·41-s − 6.48·43-s − 12.0·47-s − 6.82·49-s + 1.82·51-s + 0.171·53-s − 2·57-s − 3.58·59-s + 3.82·61-s + 0.414·63-s − 9.24·67-s − 4.82·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.156·7-s + 0.333·9-s − 0.727·11-s + 0.277·13-s + 0.443·17-s − 0.458·19-s + 0.0903·21-s − 1.00·23-s + 0.192·27-s + 0.493·29-s + 0.284·31-s − 0.420·33-s − 0.601·37-s + 0.160·39-s − 0.883·41-s − 0.988·43-s − 1.76·47-s − 0.975·49-s + 0.256·51-s + 0.0235·53-s − 0.264·57-s − 0.466·59-s + 0.490·61-s + 0.0521·63-s − 1.12·67-s − 0.581·69-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 \( 1 - 0.414T + 7T^{2} \)
11 \( 1 + 2.41T + 11T^{2} \)
17 \( 1 - 1.82T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 - 1.58T + 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 + 6.48T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 - 0.171T + 53T^{2} \)
59 \( 1 + 3.58T + 59T^{2} \)
61 \( 1 - 3.82T + 61T^{2} \)
67 \( 1 + 9.24T + 67T^{2} \)
71 \( 1 - 3.65T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + 3.17T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−7.85169013255085771952788934668, −6.74447949523242070017567938095, −6.29375670655904299077238619417, −5.25548477135529600953360069665, −4.75281748996854592338607619198, −3.76168783065216142658572618837, −3.16346027183524709894427240810, −2.23760123399759453338661676653, −1.45145032964540209968022734204, 0, 1.45145032964540209968022734204, 2.23760123399759453338661676653, 3.16346027183524709894427240810, 3.76168783065216142658572618837, 4.75281748996854592338607619198, 5.25548477135529600953360069665, 6.29375670655904299077238619417, 6.74447949523242070017567938095, 7.85169013255085771952788934668

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