Properties

Label 2-12-4.3-c12-0-0
Degree $2$
Conductor $12$
Sign $-0.998 - 0.0467i$
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  + (−46.3 + 44.1i)2-s − 420. i·3-s + (191. − 4.09e3i)4-s + 3.88e3·5-s + (1.85e4 + 1.94e4i)6-s + 5.71e4i·7-s + (1.71e5 + 1.97e5i)8-s − 1.77e5·9-s + (−1.79e5 + 1.71e5i)10-s − 6.82e5i·11-s + (−1.72e6 − 8.05e4i)12-s − 8.14e6·13-s + (−2.52e6 − 2.64e6i)14-s − 1.63e6i·15-s + (−1.67e7 − 1.56e6i)16-s − 2.04e7·17-s + ⋯
L(s)  = 1  + (−0.723 + 0.690i)2-s − 0.577i·3-s + (0.0467 − 0.998i)4-s + 0.248·5-s + (0.398 + 0.417i)6-s + 0.485i·7-s + (0.655 + 0.754i)8-s − 0.333·9-s + (−0.179 + 0.171i)10-s − 0.385i·11-s + (−0.576 − 0.0269i)12-s − 1.68·13-s + (−0.335 − 0.351i)14-s − 0.143i·15-s + (−0.995 − 0.0933i)16-s − 0.847·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0467i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.998 - 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.998 - 0.0467i$
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: \((2,\ 12,\ (\ :6),\ -0.998 - 0.0467i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.00405110 + 0.173217i\)
\(L(\frac12)\) \(\approx\) \(0.00405110 + 0.173217i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (46.3 - 44.1i)T \)
3 \( 1 + 420. iT \)
good5 \( 1 - 3.88e3T + 2.44e8T^{2} \)
7 \( 1 - 5.71e4iT - 1.38e10T^{2} \)
11 \( 1 + 6.82e5iT - 3.13e12T^{2} \)
13 \( 1 + 8.14e6T + 2.32e13T^{2} \)
17 \( 1 + 2.04e7T + 5.82e14T^{2} \)
19 \( 1 - 6.40e7iT - 2.21e15T^{2} \)
23 \( 1 - 6.19e6iT - 2.19e16T^{2} \)
29 \( 1 + 6.00e7T + 3.53e17T^{2} \)
31 \( 1 - 1.27e9iT - 7.87e17T^{2} \)
37 \( 1 + 1.65e9T + 6.58e18T^{2} \)
41 \( 1 + 5.89e9T + 2.25e19T^{2} \)
43 \( 1 + 1.66e9iT - 3.99e19T^{2} \)
47 \( 1 + 1.58e10iT - 1.16e20T^{2} \)
53 \( 1 - 4.22e10T + 4.91e20T^{2} \)
59 \( 1 + 4.29e10iT - 1.77e21T^{2} \)
61 \( 1 + 3.28e10T + 2.65e21T^{2} \)
67 \( 1 + 5.64e10iT - 8.18e21T^{2} \)
71 \( 1 - 1.97e11iT - 1.64e22T^{2} \)
73 \( 1 + 2.06e11T + 2.29e22T^{2} \)
79 \( 1 + 2.59e11iT - 5.90e22T^{2} \)
83 \( 1 - 5.86e11iT - 1.06e23T^{2} \)
89 \( 1 - 1.26e11T + 2.46e23T^{2} \)
97 \( 1 + 1.19e12T + 6.93e23T^{2} \)
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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

−17.79839036109256723220630164336, −16.72404817341878273784643925379, −15.15592215269309405917531377370, −13.89964461166095870149965195009, −12.03287080096608237034031649223, −10.10763915715240555832686080596, −8.554023151957138070019859031060, −7.04321226298516598688307788797, −5.46871419844001154686745810337, −1.98576128434009153929567056189, 0.092709979089580576132203749601, 2.41674129586190948098194027965, 4.45776464461001102178588982195, 7.31362923351001390696783416445, 9.225513640996072705089400243788, 10.30163020129003764996829016627, 11.74644127071100172953524844813, 13.39025025032420626428876300861, 15.28543644128040460081137906784, 16.92202033905555642422514668166

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