Properties

Label 2-2646-21.20-c1-0-32
Degree $2$
Conductor $2646$
Sign $-0.755 + 0.654i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s − 2.44·5-s + i·8-s + 2.44i·10-s + 4.24i·11-s + 0.717i·13-s + 16-s − 2.44·17-s + 4.89i·19-s + 2.44·20-s + 4.24·22-s − 6i·23-s + 0.999·25-s + 0.717·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 1.09·5-s + 0.353i·8-s + 0.774i·10-s + 1.27i·11-s + 0.198i·13-s + 0.250·16-s − 0.594·17-s + 1.12i·19-s + 0.547·20-s + 0.904·22-s − 1.25i·23-s + 0.199·25-s + 0.140·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.755 + 0.654i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6336031289\)
\(L(\frac12)\) \(\approx\) \(0.6336031289\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.44T + 5T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 - 0.717iT - 13T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 1.75iT - 29T^{2} \)
31 \( 1 + 9.08iT - 31T^{2} \)
37 \( 1 - 5.24T + 37T^{2} \)
41 \( 1 - 2.44T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 + 14.4iT - 53T^{2} \)
59 \( 1 - 2.44T + 59T^{2} \)
61 \( 1 + 4.18iT - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + 5.49iT - 73T^{2} \)
79 \( 1 - 0.757T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + 3.04T + 89T^{2} \)
97 \( 1 + 3.16iT - 97T^{2} \)
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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

−8.414486176313004542930674134874, −8.014582683106447552186947954706, −7.16087502938463608532203302139, −6.34306744170816537766240625883, −5.17767759794147524341159931461, −4.19422762646589139150766650826, −4.00243269800576322729836035344, −2.68535881997343551267743626001, −1.79222205244829719042190605170, −0.25908327105876428669170023508, 0.974296306875470715659204744582, 2.83078959449366713392229140092, 3.63880931653081089649976108886, 4.45849164022975878812030367028, 5.31818560401698622244649109637, 6.14702432244868915102426287867, 6.95966355722388045452383367486, 7.66666172554034912747954815708, 8.282068249244250945731497451785, 8.928737515230450971568946280004

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