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Properties

Label 2-2646-63.16-c1-0-25
Degree $2$
Conductor $2646$
Sign $0.678 + 0.734i$
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 + 3·5-s + 0.999·8-s + (−1.5 − 2.59i)10-s + 3·11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (−1.5 − 2.59i)17-s + (−3.5 + 6.06i)19-s + (−1.49 + 2.59i)20-s + (−1.5 − 2.59i)22-s + 9·23-s + 4·25-s + (−0.499 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.34·5-s + 0.353·8-s + (−0.474 − 0.821i)10-s + 0.904·11-s + (−0.138 − 0.240i)13-s + (−0.125 − 0.216i)16-s + (−0.363 − 0.630i)17-s + (−0.802 + 1.39i)19-s + (−0.335 + 0.580i)20-s + (−0.319 − 0.553i)22-s + 1.87·23-s + 0.800·25-s + (−0.0980 + 0.169i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.678 + 0.734i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.059755003\)
\(L(\frac12)\) \(\approx\) \(2.059755003\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 9T + 23T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.996795123489523937292537621138, −8.249127449229970985895755811609, −7.21931037219834221361470092867, −6.41038279175996214126638083206, −5.73490846078038708772837271758, −4.77525590125806992299219712494, −3.85077819099246777347765305773, −2.73411221058803828276500954478, −1.96783547546123729022276195676, −0.951670530352665084258329685719, 1.07132404484260927136148933198, 2.04936540956762329067166001630, 3.15130343657822466505286283482, 4.57993661655689239492855506313, 5.07578500665887372980434998798, 6.16728028892008344265010391605, 6.61800540471677300778273514118, 7.15830515947852751820671674706, 8.478800719826549423130871374049, 8.994942055761353529897793517947

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