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Properties

Label 2-2646-63.16-c1-0-8
Degree $2$
Conductor $2646$
Sign $-0.512 - 0.858i$
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 − 3.86·5-s − 0.999·8-s + (−1.93 − 3.34i)10-s + 3.73·11-s + (−3.34 − 5.79i)13-s + (−0.5 − 0.866i)16-s + (2.70 + 4.69i)17-s + (1.48 − 2.56i)19-s + (1.93 − 3.34i)20-s + (1.86 + 3.23i)22-s + 1.46·23-s + 9.92·25-s + (3.34 − 5.79i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.72·5-s − 0.353·8-s + (−0.610 − 1.05i)10-s + 1.12·11-s + (−0.928 − 1.60i)13-s + (−0.125 − 0.216i)16-s + (0.656 + 1.13i)17-s + (0.340 − 0.589i)19-s + (0.431 − 0.748i)20-s + (0.397 + 0.689i)22-s + 0.305·23-s + 1.98·25-s + (0.656 − 1.13i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.512 - 0.858i$
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: \((2,\ 2646,\ (\ :1/2),\ -0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.076868294\)
\(L(\frac12)\) \(\approx\) \(1.076868294\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.86T + 5T^{2} \)
11 \( 1 - 3.73T + 11T^{2} \)
13 \( 1 + (3.34 + 5.79i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.70 - 4.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.48 + 2.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.896 - 1.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.267 - 0.464i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.637 + 1.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.86 - 3.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.27 - 9.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.46 + 2.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.31 - 7.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.48 + 6.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.76 - 4.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 + (-3.41 - 5.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.46 - 4.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.95 - 15.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.53 + 6.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.94 + 5.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.788498008510003292421554426446, −8.172936871519732244078532373887, −7.51827413553420148137868905781, −7.07872796941963553157365842531, −6.04847382172250487330553412050, −5.14506448294195191546010051772, −4.36888104986101596169894554632, −3.56982817720446920863602739155, −2.98668187692287378261444117146, −0.979983422690407229856585845511, 0.40475952531485826816144046482, 1.75867476307007818278235439593, 3.05000108132516899128148312269, 3.85721463803705511205118420877, 4.37986887151909840230504682901, 5.14412715638283036810696334877, 6.41081942372134026387953133184, 7.23266011237203418700986895571, 7.65809286181739656480030599666, 8.807360693322438680260785916281

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