Properties

Label 2-5635-1.1-c1-0-136
Degree $2$
Conductor $5635$
Sign $1$
Analytic cond. $44.9957$
Root an. cond. $6.70788$
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  + 2·2-s + 2·4-s + 5-s − 3·9-s + 2·10-s + 2·11-s + 2·13-s − 4·16-s − 3·17-s − 6·18-s + 2·19-s + 2·20-s + 4·22-s + 23-s + 25-s + 4·26-s + 7·29-s + 5·31-s − 8·32-s − 6·34-s − 6·36-s + 11·37-s + 4·38-s − 41-s + 4·44-s − 3·45-s + 2·46-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.447·5-s − 9-s + 0.632·10-s + 0.603·11-s + 0.554·13-s − 16-s − 0.727·17-s − 1.41·18-s + 0.458·19-s + 0.447·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.784·26-s + 1.29·29-s + 0.898·31-s − 1.41·32-s − 1.02·34-s − 36-s + 1.80·37-s + 0.648·38-s − 0.156·41-s + 0.603·44-s − 0.447·45-s + 0.294·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5635\)    =    \(5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(44.9957\)
Root analytic conductor: \(6.70788\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5635} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5635,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.501230311\)
\(L(\frac12)\) \(\approx\) \(4.501230311\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 \)
23 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.263970881296066983490176011043, −6.99788152624213115553514070657, −6.45697356274813578150694386716, −5.87176551907712567991921014352, −5.27308193075008769160572382867, −4.45846692453006168600746685551, −3.82026530899319001848621226796, −2.86980424199587493895479247872, −2.38005066461420004928870391491, −0.921910412570228599339074523899, 0.921910412570228599339074523899, 2.38005066461420004928870391491, 2.86980424199587493895479247872, 3.82026530899319001848621226796, 4.45846692453006168600746685551, 5.27308193075008769160572382867, 5.87176551907712567991921014352, 6.45697356274813578150694386716, 6.99788152624213115553514070657, 8.263970881296066983490176011043

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