Properties

Label 2-1875-1.1-c1-0-70
Degree $2$
Conductor $1875$
Sign $-1$
Analytic cond. $14.9719$
Root an. cond. $3.86936$
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  + 0.536·2-s + 3-s − 1.71·4-s + 0.536·6-s + 2.57·7-s − 1.99·8-s + 9-s − 5.14·11-s − 1.71·12-s − 1.47·13-s + 1.38·14-s + 2.35·16-s + 0.687·17-s + 0.536·18-s − 8.09·19-s + 2.57·21-s − 2.75·22-s + 0.372·23-s − 1.99·24-s − 0.791·26-s + 27-s − 4.40·28-s + 0.0356·29-s − 4.48·31-s + 5.24·32-s − 5.14·33-s + 0.369·34-s + ⋯
L(s)  = 1  + 0.379·2-s + 0.577·3-s − 0.856·4-s + 0.219·6-s + 0.972·7-s − 0.704·8-s + 0.333·9-s − 1.55·11-s − 0.494·12-s − 0.409·13-s + 0.368·14-s + 0.588·16-s + 0.166·17-s + 0.126·18-s − 1.85·19-s + 0.561·21-s − 0.588·22-s + 0.0777·23-s − 0.406·24-s − 0.155·26-s + 0.192·27-s − 0.832·28-s + 0.00662·29-s − 0.806·31-s + 0.927·32-s − 0.895·33-s + 0.0632·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(14.9719\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1875,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 \)
good2 \( 1 - 0.536T + 2T^{2} \)
7 \( 1 - 2.57T + 7T^{2} \)
11 \( 1 + 5.14T + 11T^{2} \)
13 \( 1 + 1.47T + 13T^{2} \)
17 \( 1 - 0.687T + 17T^{2} \)
19 \( 1 + 8.09T + 19T^{2} \)
23 \( 1 - 0.372T + 23T^{2} \)
29 \( 1 - 0.0356T + 29T^{2} \)
31 \( 1 + 4.48T + 31T^{2} \)
37 \( 1 - 1.90T + 37T^{2} \)
41 \( 1 + 5.16T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 - 8.52T + 47T^{2} \)
53 \( 1 + 9.12T + 53T^{2} \)
59 \( 1 + 0.176T + 59T^{2} \)
61 \( 1 + 6.41T + 61T^{2} \)
67 \( 1 - 0.0834T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 - 4.95T + 79T^{2} \)
83 \( 1 - 9.36T + 83T^{2} \)
89 \( 1 + 0.0123T + 89T^{2} \)
97 \( 1 + 7.62T + 97T^{2} \)
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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

−8.638484297256271273136065478956, −8.171493157430687292406781803060, −7.55917774647914358983714229738, −6.35192272848276600694805805290, −5.23293544503174217089370391521, −4.81693562380041989020552407724, −3.93193452751325646924481011191, −2.85731840468184133440358569532, −1.86511043131658266761788788165, 0, 1.86511043131658266761788788165, 2.85731840468184133440358569532, 3.93193452751325646924481011191, 4.81693562380041989020552407724, 5.23293544503174217089370391521, 6.35192272848276600694805805290, 7.55917774647914358983714229738, 8.171493157430687292406781803060, 8.638484297256271273136065478956

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