Properties

Label 2-2646-1.1-c1-0-46
Degree $2$
Conductor $2646$
Sign $-1$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 3.64·5-s − 8-s − 3.64·10-s − 3.64·11-s − 4.64·13-s + 16-s − 3.64·17-s + 2·19-s + 3.64·20-s + 3.64·22-s + 1.29·23-s + 8.29·25-s + 4.64·26-s − 2.35·29-s − 4.64·31-s − 32-s + 3.64·34-s − 11.9·37-s − 2·38-s − 3.64·40-s − 10.9·41-s + 5·43-s − 3.64·44-s − 1.29·46-s − 4.93·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.63·5-s − 0.353·8-s − 1.15·10-s − 1.09·11-s − 1.28·13-s + 0.250·16-s − 0.884·17-s + 0.458·19-s + 0.815·20-s + 0.777·22-s + 0.269·23-s + 1.65·25-s + 0.911·26-s − 0.437·29-s − 0.834·31-s − 0.176·32-s + 0.625·34-s − 1.96·37-s − 0.324·38-s − 0.576·40-s − 1.70·41-s + 0.762·43-s − 0.549·44-s − 0.190·46-s − 0.720·47-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: no
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 \( 1 - 3.64T + 5T^{2} \)
11 \( 1 + 3.64T + 11T^{2} \)
13 \( 1 + 4.64T + 13T^{2} \)
17 \( 1 + 3.64T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 1.29T + 23T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
31 \( 1 + 4.64T + 31T^{2} \)
37 \( 1 + 11.9T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + 4.93T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 8.35T + 59T^{2} \)
61 \( 1 - 7.35T + 61T^{2} \)
67 \( 1 + 2.29T + 67T^{2} \)
71 \( 1 + 15.6T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + 4.93T + 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−8.709894022894092850540669708472, −7.68885320052820091578852436965, −7.01226476655276887425478132956, −6.28123610480342635751617364749, −5.27893624167987078723081192796, −4.98517145628389106561709250795, −3.23747365183452452773022089499, −2.30815047777987556686118546361, −1.73702203452452869574318610693, 0, 1.73702203452452869574318610693, 2.30815047777987556686118546361, 3.23747365183452452773022089499, 4.98517145628389106561709250795, 5.27893624167987078723081192796, 6.28123610480342635751617364749, 7.01226476655276887425478132956, 7.68885320052820091578852436965, 8.709894022894092850540669708472

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