Properties

Label 2-2475-1.1-c1-0-42
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
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  + 2.41·2-s + 3.82·4-s − 0.414·7-s + 4.41·8-s + 11-s + 2.82·13-s − 0.999·14-s + 2.99·16-s + 2.41·17-s + 6.41·19-s + 2.41·22-s + 23-s + 6.82·26-s − 1.58·28-s − 1.17·29-s − 8.48·31-s − 1.58·32-s + 5.82·34-s + 0.171·37-s + 15.4·38-s + 10.8·41-s + 11.6·43-s + 3.82·44-s + 2.41·46-s − 7.48·47-s − 6.82·49-s + 10.8·52-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.91·4-s − 0.156·7-s + 1.56·8-s + 0.301·11-s + 0.784·13-s − 0.267·14-s + 0.749·16-s + 0.585·17-s + 1.47·19-s + 0.514·22-s + 0.208·23-s + 1.33·26-s − 0.299·28-s − 0.217·29-s − 1.52·31-s − 0.280·32-s + 0.999·34-s + 0.0282·37-s + 2.51·38-s + 1.70·41-s + 1.77·43-s + 0.577·44-s + 0.355·46-s − 1.09·47-s − 0.975·49-s + 1.50·52-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: no
Arithmetic: yes
Character: $\chi_{2475} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.432303933\)
\(L(\frac12)\) \(\approx\) \(5.432303933\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 2T^{2} \)
7 \( 1 + 0.414T + 7T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 - 2.41T + 17T^{2} \)
19 \( 1 - 6.41T + 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + 8.48T + 31T^{2} \)
37 \( 1 - 0.171T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + 7.48T + 47T^{2} \)
53 \( 1 - 7.65T + 53T^{2} \)
59 \( 1 + 11T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 - 0.343T + 67T^{2} \)
71 \( 1 + 7.82T + 71T^{2} \)
73 \( 1 - 8.82T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + 4.48T + 83T^{2} \)
89 \( 1 + 3.65T + 89T^{2} \)
97 \( 1 + 5.82T + 97T^{2} \)
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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

−9.042108873220691970721145327688, −7.78903571165956974370921895178, −7.21133434994057695213254037416, −6.28788355886798183937497448502, −5.67279215383912692861881256720, −5.05311411854726103041484059790, −3.98889211675823804481276091346, −3.47813171042685847801851810315, −2.58177776637967524473544864204, −1.29914144667592596561329551750, 1.29914144667592596561329551750, 2.58177776637967524473544864204, 3.47813171042685847801851810315, 3.98889211675823804481276091346, 5.05311411854726103041484059790, 5.67279215383912692861881256720, 6.28788355886798183937497448502, 7.21133434994057695213254037416, 7.78903571165956974370921895178, 9.042108873220691970721145327688

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