Properties

Label 2-1875-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.59·2-s − 3-s + 4.74·4-s + 2.59·6-s − 3.28·7-s − 7.12·8-s + 9-s + 4.30·11-s − 4.74·12-s + 3.46·13-s + 8.52·14-s + 9.02·16-s + 5.44·17-s − 2.59·18-s − 7.63·19-s + 3.28·21-s − 11.1·22-s + 5.04·23-s + 7.12·24-s − 8.98·26-s − 27-s − 15.5·28-s − 3.12·29-s − 2.06·31-s − 9.18·32-s − 4.30·33-s − 14.1·34-s + ⋯
L(s)  = 1  − 1.83·2-s − 0.577·3-s + 2.37·4-s + 1.06·6-s − 1.24·7-s − 2.52·8-s + 0.333·9-s + 1.29·11-s − 1.36·12-s + 0.959·13-s + 2.27·14-s + 2.25·16-s + 1.32·17-s − 0.612·18-s − 1.75·19-s + 0.716·21-s − 2.38·22-s + 1.05·23-s + 1.45·24-s − 1.76·26-s − 0.192·27-s − 2.94·28-s − 0.579·29-s − 0.371·31-s − 1.62·32-s − 0.749·33-s − 2.42·34-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: \(0\)
Selberg data: \((2,\ 1875,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5199635846\)
\(L(\frac12)\) \(\approx\) \(0.5199635846\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.59T + 2T^{2} \)
7 \( 1 + 3.28T + 7T^{2} \)
11 \( 1 - 4.30T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 5.44T + 17T^{2} \)
19 \( 1 + 7.63T + 19T^{2} \)
23 \( 1 - 5.04T + 23T^{2} \)
29 \( 1 + 3.12T + 29T^{2} \)
31 \( 1 + 2.06T + 31T^{2} \)
37 \( 1 - 1.89T + 37T^{2} \)
41 \( 1 - 3.89T + 41T^{2} \)
43 \( 1 - 3.20T + 43T^{2} \)
47 \( 1 + 6.28T + 47T^{2} \)
53 \( 1 - 2.51T + 53T^{2} \)
59 \( 1 - 7.72T + 59T^{2} \)
61 \( 1 + 2.95T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + 4.72T + 71T^{2} \)
73 \( 1 - 1.64T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 - 5.01T + 83T^{2} \)
89 \( 1 + 9.00T + 89T^{2} \)
97 \( 1 + 2.57T + 97T^{2} \)
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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

−9.226588267639819536938246723915, −8.723914994521526785697953390100, −7.76298511569561102116459399826, −6.83839337817647391921811050301, −6.41438958673170026297480235631, −5.76906262436519212791570919478, −4.02169175278309772919120978791, −3.07524985270538143129790357045, −1.67038644175790812349741164532, −0.68220766183107318491459509569, 0.68220766183107318491459509569, 1.67038644175790812349741164532, 3.07524985270538143129790357045, 4.02169175278309772919120978791, 5.76906262436519212791570919478, 6.41438958673170026297480235631, 6.83839337817647391921811050301, 7.76298511569561102116459399826, 8.723914994521526785697953390100, 9.226588267639819536938246723915

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