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Properties

Label 2-2646-63.4-c1-0-9
Degree $2$
Conductor $2646$
Sign $-0.910 - 0.412i$
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 + 1.58·5-s + 0.999·8-s + (−0.794 + 1.37i)10-s + 1.58·11-s + (−2.40 + 4.16i)13-s + (−0.5 + 0.866i)16-s + (−2.69 + 4.67i)17-s + (3.54 + 6.14i)19-s + (−0.794 − 1.37i)20-s + (−0.794 + 1.37i)22-s − 0.300·23-s − 2.47·25-s + (−2.40 − 4.16i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.710·5-s + 0.353·8-s + (−0.251 + 0.434i)10-s + 0.478·11-s + (−0.667 + 1.15i)13-s + (−0.125 + 0.216i)16-s + (−0.654 + 1.13i)17-s + (0.814 + 1.41i)19-s + (−0.177 − 0.307i)20-s + (−0.169 + 0.293i)22-s − 0.0626·23-s − 0.495·25-s + (−0.471 − 0.817i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.910 - 0.412i$
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: \((2,\ 2646,\ (\ :1/2),\ -0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.044210504\)
\(L(\frac12)\) \(\approx\) \(1.044210504\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.58T + 5T^{2} \)
11 \( 1 - 1.58T + 11T^{2} \)
13 \( 1 + (2.40 - 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.69 - 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.54 - 6.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.300T + 23T^{2} \)
29 \( 1 + (4.13 + 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.35 + 2.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.33 - 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.44 - 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.23 + 5.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.23 - 3.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.02 - 8.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (8.02 - 13.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.19 - 7.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.18 - 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.60 - 2.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.712 + 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.262006779372851732478694948134, −8.425346178158584073158951986391, −7.63355965721110419908354141361, −6.92526758564507341761471982240, −5.96915596694431910646202068630, −5.75571328522432190000403683094, −4.43725337664072724268039868520, −3.83260677269417275106045533760, −2.22422819581699282909599387864, −1.51204809033762697135603639441, 0.37188822306373035435621985799, 1.64319936287207927053557051848, 2.71205462837721766423538016649, 3.32028113670895420689289483959, 4.71779462899036175114494863800, 5.20159700223464074018971550259, 6.24562244272745633764576615040, 7.22146798694265698512508677895, 7.68916843317197173952665717988, 8.979990157705434628812976139966

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