Properties

Label 2-2646-1.1-c1-0-50
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 6·11-s − 5·13-s + 16-s + 6·17-s + 4·19-s − 6·22-s − 6·23-s − 5·25-s − 5·26-s − 6·29-s + 31-s + 32-s + 6·34-s − 37-s + 4·38-s − 6·41-s − 43-s − 6·44-s − 6·46-s − 6·47-s − 5·50-s − 5·52-s + 6·53-s − 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s − 1.38·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 1.27·22-s − 1.25·23-s − 25-s − 0.980·26-s − 1.11·29-s + 0.179·31-s + 0.176·32-s + 1.02·34-s − 0.164·37-s + 0.648·38-s − 0.937·41-s − 0.152·43-s − 0.904·44-s − 0.884·46-s − 0.875·47-s − 0.707·50-s − 0.693·52-s + 0.824·53-s − 0.787·58-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: \(1\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−8.018373233993352526075351795604, −7.76500604494403870217253478491, −7.08399389581964304580937840750, −5.76516762621587586677817121204, −5.43294840515298920841807654421, −4.67577382905622175502731044741, −3.54405748811653734427126185430, −2.77713768583350957199647900568, −1.85393606295065766044693801451, 0, 1.85393606295065766044693801451, 2.77713768583350957199647900568, 3.54405748811653734427126185430, 4.67577382905622175502731044741, 5.43294840515298920841807654421, 5.76516762621587586677817121204, 7.08399389581964304580937840750, 7.76500604494403870217253478491, 8.018373233993352526075351795604

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