Properties

Label 2-2646-1.1-c1-0-5
Degree $2$
Conductor $2646$
Sign $1$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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 − 2·5-s − 8-s + 2·10-s + 2·11-s + 13-s + 16-s + 3·17-s + 4·19-s − 2·20-s − 2·22-s − 3·23-s − 25-s − 26-s − 7·29-s + 11·31-s − 32-s − 3·34-s − 6·37-s − 4·38-s + 2·40-s − 6·41-s + 43-s + 2·44-s + 3·46-s − 8·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 0.603·11-s + 0.277·13-s + 1/4·16-s + 0.727·17-s + 0.917·19-s − 0.447·20-s − 0.426·22-s − 0.625·23-s − 1/5·25-s − 0.196·26-s − 1.29·29-s + 1.97·31-s − 0.176·32-s − 0.514·34-s − 0.986·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s + 0.152·43-s + 0.301·44-s + 0.442·46-s − 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2646} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.051930162\)
\(L(\frac12)\) \(\approx\) \(1.051930162\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−8.692207491505439003595372266413, −8.149259474632010208639933009664, −7.50081794391536701476109235300, −6.77206025715325106952112089107, −5.92058744841100945848362663445, −4.96231199029695381053928434268, −3.82219360233253453090009490035, −3.27085750102907089905785288191, −1.88945277610592292237661176672, −0.72482106189716605193481872454, 0.72482106189716605193481872454, 1.88945277610592292237661176672, 3.27085750102907089905785288191, 3.82219360233253453090009490035, 4.96231199029695381053928434268, 5.92058744841100945848362663445, 6.77206025715325106952112089107, 7.50081794391536701476109235300, 8.149259474632010208639933009664, 8.692207491505439003595372266413

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