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Properties

Label 2-12-4.3-c12-0-3
Degree $2$
Conductor $12$
Sign $0.544 - 0.838i$
Analytic cond. $10.9679$
Root an. cond. $3.31178$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (18.1 − 61.3i)2-s + 420. i·3-s + (−3.43e3 − 2.23e3i)4-s − 204.·5-s + (2.58e4 + 7.65e3i)6-s + 1.21e5i·7-s + (−1.99e5 + 1.70e5i)8-s − 1.77e5·9-s + (−3.71e3 + 1.25e4i)10-s + 2.28e6i·11-s + (9.38e5 − 1.44e6i)12-s + 6.03e6·13-s + (7.46e6 + 2.21e6i)14-s − 8.60e4i·15-s + (6.82e6 + 1.53e7i)16-s − 1.11e7·17-s + ⋯
L(s)  = 1  + (0.284 − 0.958i)2-s + 0.577i·3-s + (−0.838 − 0.544i)4-s − 0.0130·5-s + (0.553 + 0.163i)6-s + 1.03i·7-s + (−0.760 + 0.649i)8-s − 0.333·9-s + (−0.00371 + 0.0125i)10-s + 1.29i·11-s + (0.314 − 0.484i)12-s + 1.24·13-s + (0.991 + 0.293i)14-s − 0.00755i·15-s + (0.406 + 0.913i)16-s − 0.459·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.544 - 0.838i$
Analytic conductor: \(10.9679\)
Root analytic conductor: \(3.31178\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :6),\ 0.544 - 0.838i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.21081 + 0.657396i\)
\(L(\frac12)\) \(\approx\) \(1.21081 + 0.657396i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-18.1 + 61.3i)T \)
3 \( 1 - 420. iT \)
good5 \( 1 + 204.T + 2.44e8T^{2} \)
7 \( 1 - 1.21e5iT - 1.38e10T^{2} \)
11 \( 1 - 2.28e6iT - 3.13e12T^{2} \)
13 \( 1 - 6.03e6T + 2.32e13T^{2} \)
17 \( 1 + 1.11e7T + 5.82e14T^{2} \)
19 \( 1 - 5.58e7iT - 2.21e15T^{2} \)
23 \( 1 + 7.22e7iT - 2.19e16T^{2} \)
29 \( 1 + 7.70e8T + 3.53e17T^{2} \)
31 \( 1 - 3.21e8iT - 7.87e17T^{2} \)
37 \( 1 + 3.35e9T + 6.58e18T^{2} \)
41 \( 1 - 6.76e9T + 2.25e19T^{2} \)
43 \( 1 - 3.68e9iT - 3.99e19T^{2} \)
47 \( 1 + 7.64e9iT - 1.16e20T^{2} \)
53 \( 1 - 2.39e10T + 4.91e20T^{2} \)
59 \( 1 + 6.16e10iT - 1.77e21T^{2} \)
61 \( 1 - 8.37e10T + 2.65e21T^{2} \)
67 \( 1 - 1.69e11iT - 8.18e21T^{2} \)
71 \( 1 - 1.22e11iT - 1.64e22T^{2} \)
73 \( 1 + 1.26e11T + 2.29e22T^{2} \)
79 \( 1 + 3.45e10iT - 5.90e22T^{2} \)
83 \( 1 + 2.27e11iT - 1.06e23T^{2} \)
89 \( 1 + 7.45e11T + 2.46e23T^{2} \)
97 \( 1 - 1.04e12T + 6.93e23T^{2} \)
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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

−17.75416881194216512537173929109, −15.70523418278996592095396741235, −14.54167574337400936299189490887, −12.82205971428195458504249198987, −11.55881506292851018293414947061, −10.06007585837038815678218377906, −8.742461304270461771127594967501, −5.64565545645210864581507177346, −3.92894857835395170367161808835, −2.00796309419130781804506261026, 0.58570228701622891830256652540, 3.74432113706509519101254835246, 5.92081732381062836489849017978, 7.35581997111206688881884029605, 8.805663290932218951853982449279, 11.18247524506046164157379058973, 13.30858750426571472950151357568, 13.82379509027578550690990012036, 15.66509148055617887596867498671, 16.88234484218738008166638202786

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