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Properties

Label 2-2646-63.16-c1-0-20
Degree $2$
Conductor $2646$
Sign $0.0977 - 0.995i$
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 + 3.69·5-s − 0.999·8-s + (1.84 + 3.20i)10-s + 1.47·11-s + (1.34 + 2.33i)13-s + (−0.5 − 0.866i)16-s + (3.28 + 5.69i)17-s + (0.444 − 0.769i)19-s + (−1.84 + 3.20i)20-s + (0.738 + 1.27i)22-s − 6.28·23-s + 8.68·25-s + (−1.34 + 2.33i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.65·5-s − 0.353·8-s + (0.584 + 1.01i)10-s + 0.445·11-s + (0.374 + 0.648i)13-s + (−0.125 − 0.216i)16-s + (0.797 + 1.38i)17-s + (0.101 − 0.176i)19-s + (−0.413 + 0.716i)20-s + (0.157 + 0.272i)22-s − 1.31·23-s + 1.73·25-s + (−0.264 + 0.458i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.0977 - 0.995i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.078019664\)
\(L(\frac12)\) \(\approx\) \(3.078019664\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.69T + 5T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + (-1.34 - 2.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.28 - 5.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.444 + 0.769i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.28T + 23T^{2} \)
29 \( 1 + (1.25 - 2.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.40 + 5.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.38 - 2.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.05 + 3.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.00618 + 0.0107i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.49 - 6.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.60 - 2.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.45 - 5.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.86 + 4.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.73 + 8.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 + (-6.03 - 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.72 + 9.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.23 + 3.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.43 - 7.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.58 + 11.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.011999729054147261221431081230, −8.311775308019476765130553407384, −7.41597980940053283312515669444, −6.28167330443988820580573923550, −6.17906056123312900156954257223, −5.40329939298242189544671305129, −4.38182416356089857901573434537, −3.53026728226374695063446698909, −2.26389189478395841312066142178, −1.42709510953193061549702397041, 0.956690062478187076041672127339, 1.94587507594735887258855661482, 2.79860340199975724241502689734, 3.69392261832816217191935521782, 4.91754423902318198144283100496, 5.52088050938005927959185268565, 6.14087023901194480402625706079, 6.93891824624153878713698858160, 8.054436583821871168205594052531, 8.949257410437999488276420586089

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