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Properties

Label 2-2646-63.59-c1-0-12
Degree $2$
Conductor $2646$
Sign $0.487 - 0.873i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + 2.34·5-s − 0.999i·8-s + (−2.03 − 1.17i)10-s + 5.67i·11-s + (1.48 + 0.859i)13-s + (−0.5 + 0.866i)16-s + (−0.884 + 1.53i)17-s + (0.986 − 0.569i)19-s + (1.17 + 2.03i)20-s + (2.83 − 4.91i)22-s − 3.67i·23-s + 0.519·25-s + (−0.859 − 1.48i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 1.05·5-s − 0.353i·8-s + (−0.643 − 0.371i)10-s + 1.71i·11-s + (0.413 + 0.238i)13-s + (−0.125 + 0.216i)16-s + (−0.214 + 0.371i)17-s + (0.226 − 0.130i)19-s + (0.262 + 0.454i)20-s + (0.605 − 1.04i)22-s − 0.766i·23-s + 0.103·25-s + (−0.168 − 0.292i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.487 - 0.873i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.487 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.472477618\)
\(L(\frac12)\) \(\approx\) \(1.472477618\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2.34T + 5T^{2} \)
11 \( 1 - 5.67iT - 11T^{2} \)
13 \( 1 + (-1.48 - 0.859i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.884 - 1.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.986 + 0.569i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.67iT - 23T^{2} \)
29 \( 1 + (3.59 - 2.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.24 + 4.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.59 - 7.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.99 - 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.76 - 3.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.90 - 10.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.11 + 1.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.79 + 4.49i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.43 + 9.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.52iT - 71T^{2} \)
73 \( 1 + (-4.62 - 2.67i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.51 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.27 - 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.580 + 1.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.97 - 2.29i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.415981041402020613622765908246, −8.222987687223260034921398206117, −7.66003442582824294390953271182, −6.52026240502085472204104496603, −6.28294602149514187568531765216, −4.92511113712910138759455360583, −4.32608258677043269772703955844, −2.97795556932084115276095821042, −2.08891464709159822652868032868, −1.35585076202102141287446793057, 0.60089812775783458065035673344, 1.74331042185146355038645039328, 2.83980429301498225417751059513, 3.79655830297743552609912109233, 5.24196283355242386598171120852, 5.76973136046937721430288340457, 6.31329182602841877031968457165, 7.23977249704910813890814660631, 8.116308115225450470640828691327, 8.787863074957661519087110215996

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