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Properties

Label 2-12-4.3-c14-0-4
Degree $2$
Conductor $12$
Sign $0.693 - 0.720i$
Analytic cond. $14.9194$
Root an. cond. $3.86257$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−47.8 + 118. i)2-s − 1.26e3i·3-s + (−1.18e4 − 1.13e4i)4-s − 1.32e5·5-s + (1.49e5 + 6.04e4i)6-s + 3.04e5i·7-s + (1.91e6 − 8.57e5i)8-s − 1.59e6·9-s + (6.33e6 − 1.57e7i)10-s − 4.87e6i·11-s + (−1.43e7 + 1.49e7i)12-s + 9.89e7·13-s + (−3.62e7 − 1.45e7i)14-s + 1.67e8i·15-s + (1.02e7 + 2.68e8i)16-s + 4.23e7·17-s + ⋯
L(s)  = 1  + (−0.373 + 0.927i)2-s − 0.577i·3-s + (−0.720 − 0.693i)4-s − 1.69·5-s + (0.535 + 0.215i)6-s + 0.370i·7-s + (0.912 − 0.408i)8-s − 0.333·9-s + (0.633 − 1.57i)10-s − 0.250i·11-s + (−0.400 + 0.415i)12-s + 1.57·13-s + (−0.343 − 0.138i)14-s + 0.977i·15-s + (0.0380 + 0.999i)16-s + 0.103·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.693 - 0.720i$
Analytic conductor: \(14.9194\)
Root analytic conductor: \(3.86257\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :7),\ 0.693 - 0.720i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.845672 + 0.359765i\)
\(L(\frac12)\) \(\approx\) \(0.845672 + 0.359765i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (47.8 - 118. i)T \)
3 \( 1 + 1.26e3iT \)
good5 \( 1 + 1.32e5T + 6.10e9T^{2} \)
7 \( 1 - 3.04e5iT - 6.78e11T^{2} \)
11 \( 1 + 4.87e6iT - 3.79e14T^{2} \)
13 \( 1 - 9.89e7T + 3.93e15T^{2} \)
17 \( 1 - 4.23e7T + 1.68e17T^{2} \)
19 \( 1 - 1.40e9iT - 7.99e17T^{2} \)
23 \( 1 + 4.23e9iT - 1.15e19T^{2} \)
29 \( 1 + 5.57e9T + 2.97e20T^{2} \)
31 \( 1 + 9.56e8iT - 7.56e20T^{2} \)
37 \( 1 - 1.19e11T + 9.01e21T^{2} \)
41 \( 1 - 2.10e11T + 3.79e22T^{2} \)
43 \( 1 + 3.76e11iT - 7.38e22T^{2} \)
47 \( 1 - 8.87e11iT - 2.56e23T^{2} \)
53 \( 1 + 1.43e11T + 1.37e24T^{2} \)
59 \( 1 + 2.40e12iT - 6.19e24T^{2} \)
61 \( 1 + 2.37e12T + 9.87e24T^{2} \)
67 \( 1 + 1.58e12iT - 3.67e25T^{2} \)
71 \( 1 - 1.62e13iT - 8.27e25T^{2} \)
73 \( 1 - 8.92e12T + 1.22e26T^{2} \)
79 \( 1 - 1.91e13iT - 3.68e26T^{2} \)
83 \( 1 + 4.14e11iT - 7.36e26T^{2} \)
89 \( 1 - 3.16e13T + 1.95e27T^{2} \)
97 \( 1 - 6.43e13T + 6.52e27T^{2} \)
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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

−16.53345535730815229317338748492, −15.65339723799547463646874049006, −14.34973263076601499104617582258, −12.58120634201050955585825108210, −11.01545365810899623090732799014, −8.591541516588127163370555122665, −7.74835442730650371689919437572, −6.12147799465557355519144121118, −3.94455565056224553200235615904, −0.828582563106109730248591414011, 0.70180695971945335582345905618, 3.35912583917873494346698816902, 4.38371131798220838574843676387, 7.66585574825854747476886033719, 9.018181372351954511155275056442, 10.90090231239894316842637765548, 11.61333148075621406495749816963, 13.31917547330312155082735556672, 15.32381317734953157671567523858, 16.42647125866840167105345010418

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