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Properties

Label 2-2646-63.4-c1-0-4
Degree $2$
Conductor $2646$
Sign $0.0644 - 0.997i$
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 + 1.76·5-s − 0.999·8-s + (0.880 − 1.52i)10-s − 6.12·11-s + (0.380 − 0.658i)13-s + (−0.5 + 0.866i)16-s + (−3.42 + 5.92i)17-s + (−0.971 − 1.68i)19-s + (−0.880 − 1.52i)20-s + (−3.06 + 5.30i)22-s + 0.421·23-s − 1.89·25-s + (−0.380 − 0.658i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.787·5-s − 0.353·8-s + (0.278 − 0.482i)10-s − 1.84·11-s + (0.105 − 0.182i)13-s + (−0.125 + 0.216i)16-s + (−0.829 + 1.43i)17-s + (−0.222 − 0.385i)19-s + (−0.196 − 0.340i)20-s + (−0.652 + 1.13i)22-s + 0.0877·23-s − 0.379·25-s + (−0.0746 − 0.129i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0644 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0644 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.0644 - 0.997i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.0644 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8038001936\)
\(L(\frac12)\) \(\approx\) \(0.8038001936\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.76T + 5T^{2} \)
11 \( 1 + 6.12T + 11T^{2} \)
13 \( 1 + (-0.380 + 0.658i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.42 - 5.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.971 + 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.421T + 23T^{2} \)
29 \( 1 + (0.732 + 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.85 - 6.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.47 - 6.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.33 - 7.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.830 - 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.112 + 0.195i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.993 + 1.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.17 - 8.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.39 + 5.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (0.153 - 0.265i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.56 + 2.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.30 - 2.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.81 - 3.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.130687196667381736857241067761, −8.341673101223437401569132167570, −7.67490362861956779645038507744, −6.42698005246277736922050001560, −5.92476795077510001837537858633, −5.03455649086500782046206191257, −4.39895763694553797256286418995, −3.13207312624467507636956722996, −2.44347804178324953485902333107, −1.50141574611203974581791229245, 0.21010193264107049735090376633, 2.16673592917480656095787711524, 2.77838004949234959925705862453, 4.05540813532629377504447923372, 5.00955430926273881807727458008, 5.51958592031001052919875076072, 6.26456402370743267608351919286, 7.20324073701225024285113055063, 7.74554418460749418744517525153, 8.610174601845248220429652012306

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