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Properties

Label 2-2646-63.4-c1-0-29
Degree $2$
Conductor $2646$
Sign $0.920 + 0.389i$
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 + 0.613·5-s + 0.999·8-s + (−0.306 + 0.531i)10-s + 4.62·11-s + (3.25 − 5.64i)13-s + (−0.5 + 0.866i)16-s + (1.01 − 1.75i)17-s + (1.32 + 2.29i)19-s + (−0.306 − 0.531i)20-s + (−2.31 + 4.00i)22-s + 3.62·23-s − 4.62·25-s + (3.25 + 5.64i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.274·5-s + 0.353·8-s + (−0.0970 + 0.168i)10-s + 1.39·11-s + (0.903 − 1.56i)13-s + (−0.125 + 0.216i)16-s + (0.246 − 0.426i)17-s + (0.303 + 0.525i)19-s + (−0.0686 − 0.118i)20-s + (−0.492 + 0.853i)22-s + 0.755·23-s − 0.924·25-s + (0.639 + 1.10i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.920 + 0.389i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.920 + 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.663827267\)
\(L(\frac12)\) \(\approx\) \(1.663827267\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.613T + 5T^{2} \)
11 \( 1 - 4.62T + 11T^{2} \)
13 \( 1 + (-3.25 + 5.64i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.01 + 1.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.32 - 2.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.25 + 5.64i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.01 + 1.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.31 - 2.27i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.29 - 9.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.11 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.56 + 6.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.31 + 4.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.376T + 71T^{2} \)
73 \( 1 + (-3.66 + 6.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.81 - 11.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.87 + 6.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.13 - 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.14 + 14.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.848665397805657786736737690634, −7.893611760253580547978444135613, −7.50351939613765285588266846684, −6.32441440680038382479045725561, −5.95765114468880912608899704583, −5.14334976979445399452539260264, −3.99039225497066192826135998347, −3.24382434724828319399101704310, −1.76157337641372659656382962143, −0.69154218078514815447702491241, 1.28896361913515283087969458896, 1.85759244927849902892385370747, 3.29451252065673083100598323492, 3.92294298912513117365102672706, 4.79461261030937088916512874856, 5.92311261036561230000277303813, 6.74839934621842932296672776242, 7.26326550969330322831295697319, 8.665258480221388222701796589155, 8.835532413373746200436300458040

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