Properties

Label 2-2646-63.16-c1-0-2
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: $\chi_{2646} (667, \cdot )$
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

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

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