Properties

Label 2-7500-1.1-c1-0-75
Degree $2$
Conductor $7500$
Sign $-1$
Analytic cond. $59.8878$
Root an. cond. $7.73872$
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 + 0.547·7-s + 9-s − 1.33·11-s + 0.302·13-s + 4.04·17-s − 5.63·19-s + 0.547·21-s + 0.245·23-s + 27-s − 1.36·29-s − 3.53·31-s − 1.33·33-s − 2.18·37-s + 0.302·39-s − 9.80·41-s − 8.35·43-s − 10.4·47-s − 6.70·49-s + 4.04·51-s − 7.69·53-s − 5.63·57-s + 4.15·59-s + 2.07·61-s + 0.547·63-s + 8.48·67-s + 0.245·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.206·7-s + 0.333·9-s − 0.403·11-s + 0.0837·13-s + 0.980·17-s − 1.29·19-s + 0.119·21-s + 0.0511·23-s + 0.192·27-s − 0.254·29-s − 0.635·31-s − 0.232·33-s − 0.359·37-s + 0.0483·39-s − 1.53·41-s − 1.27·43-s − 1.52·47-s − 0.957·49-s + 0.565·51-s − 1.05·53-s − 0.746·57-s + 0.540·59-s + 0.265·61-s + 0.0689·63-s + 1.03·67-s + 0.0295·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7500\)    =    \(2^{2} \cdot 3 \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(59.8878\)
Root analytic conductor: \(7.73872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7500,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good7 \( 1 - 0.547T + 7T^{2} \)
11 \( 1 + 1.33T + 11T^{2} \)
13 \( 1 - 0.302T + 13T^{2} \)
17 \( 1 - 4.04T + 17T^{2} \)
19 \( 1 + 5.63T + 19T^{2} \)
23 \( 1 - 0.245T + 23T^{2} \)
29 \( 1 + 1.36T + 29T^{2} \)
31 \( 1 + 3.53T + 31T^{2} \)
37 \( 1 + 2.18T + 37T^{2} \)
41 \( 1 + 9.80T + 41T^{2} \)
43 \( 1 + 8.35T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + 7.69T + 53T^{2} \)
59 \( 1 - 4.15T + 59T^{2} \)
61 \( 1 - 2.07T + 61T^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 + 9.18T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + 6.65T + 83T^{2} \)
89 \( 1 - 4.97T + 89T^{2} \)
97 \( 1 - 4.06T + 97T^{2} \)
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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

−7.74950357351917088572400329974, −6.81791847004640703997321659416, −6.31707463671146314807563963797, −5.24464202090665968031277015114, −4.81887921756291012121314941907, −3.71370611834180232909411823144, −3.26228533073044497026202779473, −2.18639411239472032429066874157, −1.49525599362996742127685520821, 0, 1.49525599362996742127685520821, 2.18639411239472032429066874157, 3.26228533073044497026202779473, 3.71370611834180232909411823144, 4.81887921756291012121314941907, 5.24464202090665968031277015114, 6.31707463671146314807563963797, 6.81791847004640703997321659416, 7.74950357351917088572400329974

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