Properties

Label 2-2646-63.4-c1-0-11
Degree $2$
Conductor $2646$
Sign $0.537 - 0.843i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 3.25·5-s + 0.999·8-s + (1.62 − 2.82i)10-s − 5.62·11-s + (−0.613 + 1.06i)13-s + (−0.5 + 0.866i)16-s + (2.95 − 5.11i)17-s + (1.32 + 2.29i)19-s + (1.62 + 2.82i)20-s + (2.81 − 4.87i)22-s − 6.62·23-s + 5.62·25-s + (−0.613 − 1.06i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.45·5-s + 0.353·8-s + (0.515 − 0.892i)10-s − 1.69·11-s + (−0.170 + 0.294i)13-s + (−0.125 + 0.216i)16-s + (0.716 − 1.24i)17-s + (0.303 + 0.525i)19-s + (0.364 + 0.631i)20-s + (0.599 − 1.03i)22-s − 1.38·23-s + 1.12·25-s + (−0.120 − 0.208i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.537 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5323087679\)
\(L(\frac12)\) \(\approx\) \(0.5323087679\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3.25T + 5T^{2} \)
11 \( 1 + 5.62T + 11T^{2} \)
13 \( 1 + (0.613 - 1.06i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.95 + 5.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.32 - 2.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.62T + 23T^{2} \)
29 \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.613 - 1.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.95 + 5.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.81 + 6.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.29 - 9.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.43 - 12.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.24 - 3.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.81 - 4.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + (-5.59 + 9.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.68 - 2.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.87 - 6.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.48 + 7.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.53 - 2.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.747609596209464497814358751990, −7.85596495536696293027516089140, −7.71728167352160901286713190755, −7.05132334325713742323490438619, −5.82040515146330592433419625298, −5.17881848252132271406876538820, −4.30062265870268739331714421963, −3.43027974537344392471221035292, −2.31974166491121168735948560706, −0.53494906639995368768937372957, 0.39871151090303408056122523740, 1.95029454977668915389949119229, 3.14307586919654715347231793409, 3.65486542283708923995104698270, 4.69336923447412401732888052331, 5.39901961928618395465836528664, 6.62342380066379208908137217003, 7.63393024612073120590359478128, 8.186994397004074175414922853615, 8.290002705128006500806014965649

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