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Properties

Label 2-2646-63.4-c1-0-30
Degree $2$
Conductor $2646$
Sign $0.143 + 0.989i$
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 + 3.44·5-s − 0.999·8-s + (1.72 − 2.98i)10-s − 4·11-s + (2.12 − 3.67i)13-s + (−0.5 + 0.866i)16-s + (0.707 − 1.22i)17-s + (3.13 + 5.43i)19-s + (−1.72 − 2.98i)20-s + (−2 + 3.46i)22-s + 8.74·23-s + 6.87·25-s + (−2.12 − 3.67i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.54·5-s − 0.353·8-s + (0.544 − 0.943i)10-s − 1.20·11-s + (0.588 − 1.01i)13-s + (−0.125 + 0.216i)16-s + (0.171 − 0.297i)17-s + (0.719 + 1.24i)19-s + (−0.385 − 0.667i)20-s + (−0.426 + 0.738i)22-s + 1.82·23-s + 1.37·25-s + (−0.416 − 0.720i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.143 + 0.989i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.143 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.841589952\)
\(L(\frac12)\) \(\approx\) \(2.841589952\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.44T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (-2.12 + 3.67i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.13 - 5.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.74T + 23T^{2} \)
29 \( 1 + (-0.563 - 0.976i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.73 + 4.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.87 + 4.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.82 + 4.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.563 + 0.976i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.03 - 3.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.30 + 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.15 + 7.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.13 - 5.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.43 - 5.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.87T + 71T^{2} \)
73 \( 1 + (2.21 - 3.82i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.936 + 1.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.32 + 2.29i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.53 - 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.50 + 13.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.911977830421063614165636664673, −8.000090139914153657571963281933, −7.11360618532690189546354621250, −6.00834114826278374104734695232, −5.43951495093389807568074129259, −5.09652974440007857806035405623, −3.61112697714350533891411453262, −2.83576080264526470913519780031, −2.00519502610533766731331240885, −0.903031220471298887667788815760, 1.28997500538607383898207090920, 2.49819066563800663626786018762, 3.24034337973634808640918188061, 4.72062631405974240701039248261, 5.15464457466950265448705815383, 5.92830359966884210911260827357, 6.69921690359707739935934165965, 7.24288767773574112369136403302, 8.319058477377355086749692645094, 9.138442914047879783815479707275

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