Properties

Label 2-2646-1.1-c1-0-11
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 − 5·11-s + 6·13-s + 16-s + 4·17-s − 4·19-s + 2·20-s + 5·22-s + 4·23-s − 25-s − 6·26-s − 7·29-s + 3·31-s − 32-s − 4·34-s + 8·37-s + 4·38-s − 2·40-s + 6·41-s + 8·43-s − 5·44-s − 4·46-s − 6·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s − 1.50·11-s + 1.66·13-s + 1/4·16-s + 0.970·17-s − 0.917·19-s + 0.447·20-s + 1.06·22-s + 0.834·23-s − 1/5·25-s − 1.17·26-s − 1.29·29-s + 0.538·31-s − 0.176·32-s − 0.685·34-s + 1.31·37-s + 0.648·38-s − 0.316·40-s + 0.937·41-s + 1.21·43-s − 0.753·44-s − 0.589·46-s − 0.875·47-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

\(L(1)\) \(\approx\) \(1.542421821\)
\(L(\frac12)\) \(\approx\) \(1.542421821\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−8.916206844022177856631722653395, −8.068147584694120529877202336385, −7.62996171509627626414725437254, −6.46036780540424052038469573050, −5.87635293671247234260736302342, −5.25718839816604212664890932158, −3.94766098556745655416613454733, −2.87326925572822337756196679887, −2.02203535562236174351419355432, −0.882876888480485871066843660929, 0.882876888480485871066843660929, 2.02203535562236174351419355432, 2.87326925572822337756196679887, 3.94766098556745655416613454733, 5.25718839816604212664890932158, 5.87635293671247234260736302342, 6.46036780540424052038469573050, 7.62996171509627626414725437254, 8.068147584694120529877202336385, 8.916206844022177856631722653395

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