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Properties

Label 2-2646-63.4-c1-0-17
Degree $2$
Conductor $2646$
Sign $0.831 + 0.554i$
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 − 3.18·5-s + 0.999·8-s + (1.59 − 2.75i)10-s − 3.18·11-s + (−2.85 + 4.93i)13-s + (−0.5 + 0.866i)16-s + (−0.760 + 1.31i)17-s + (0.641 + 1.11i)19-s + (1.59 + 2.75i)20-s + (1.59 − 2.75i)22-s − 2.23·23-s + 5.12·25-s + (−2.85 − 4.93i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.42·5-s + 0.353·8-s + (0.503 − 0.871i)10-s − 0.959·11-s + (−0.790 + 1.36i)13-s + (−0.125 + 0.216i)16-s + (−0.184 + 0.319i)17-s + (0.147 + 0.254i)19-s + (0.355 + 0.616i)20-s + (0.339 − 0.587i)22-s − 0.466·23-s + 1.02·25-s + (−0.559 − 0.968i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.831 + 0.554i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.831 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3389612216\)
\(L(\frac12)\) \(\approx\) \(0.3389612216\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.18T + 5T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
13 \( 1 + (2.85 - 4.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.760 - 1.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.641 - 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.23T + 23T^{2} \)
29 \( 1 + (-3.54 - 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.71 + 8.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.02 - 1.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.562 - 0.974i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.56 + 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.48 + 9.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.69T + 71T^{2} \)
73 \( 1 + (-2.48 + 4.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.06 + 3.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.03 + 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.112 - 0.195i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.42 + 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.623994988741571189472076544247, −7.86453506399553109595055245051, −7.45474842852149164922660666420, −6.73037940901259963244379778293, −5.78254439363188828428069777656, −4.71383326266261165764143884865, −4.26062710464139396347368206101, −3.17120428188905476849618366368, −1.88913144548316399530390403789, −0.20147753699830051876794276380, 0.68223168755965770565846557042, 2.42045039607319416795710394791, 3.12820886430964922037372299198, 4.00127049199699868644782249842, 4.87679592062663476605062096929, 5.61929337299538755113011175085, 7.08139662374109657281127776112, 7.56876014660706277319458041535, 8.169253615967829571973361462078, 8.779127240378728089597193349798

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