Properties

Label 2-2475-1.1-c1-0-27
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 3·7-s − 3·8-s + 11-s − 2·13-s + 3·14-s − 16-s + 3·17-s − 19-s + 22-s + 23-s − 2·26-s − 3·28-s + 6·29-s + 4·31-s + 5·32-s + 3·34-s − 37-s − 38-s − 5·41-s − 4·43-s − 44-s + 46-s + 3·47-s + 2·49-s + 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.13·7-s − 1.06·8-s + 0.301·11-s − 0.554·13-s + 0.801·14-s − 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.213·22-s + 0.208·23-s − 0.392·26-s − 0.566·28-s + 1.11·29-s + 0.718·31-s + 0.883·32-s + 0.514·34-s − 0.164·37-s − 0.162·38-s − 0.780·41-s − 0.609·43-s − 0.150·44-s + 0.147·46-s + 0.437·47-s + 2/7·49-s + 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.413137742\)
\(L(\frac12)\) \(\approx\) \(2.413137742\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−8.629520496333900301508791948199, −8.402697423815820371368214204898, −7.37873719468171681835668917902, −6.49977913721194341492913045454, −5.50990223518097283261703450394, −4.95186878454497229266782270060, −4.28337792928749479562771592498, −3.37396408345035735462740571382, −2.29430503516037708243219139184, −0.927224326552216153753341436475, 0.927224326552216153753341436475, 2.29430503516037708243219139184, 3.37396408345035735462740571382, 4.28337792928749479562771592498, 4.95186878454497229266782270060, 5.50990223518097283261703450394, 6.49977913721194341492913045454, 7.37873719468171681835668917902, 8.402697423815820371368214204898, 8.629520496333900301508791948199

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