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Properties

Label 2-12-4.3-c14-0-7
Degree $2$
Conductor $12$
Sign $0.539 + 0.842i$
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  + (−122. + 35.9i)2-s − 1.26e3i·3-s + (1.37e4 − 8.83e3i)4-s + 4.98e4·5-s + (4.54e4 + 1.55e5i)6-s + 9.17e4i·7-s + (−1.37e6 + 1.58e6i)8-s − 1.59e6·9-s + (−6.11e6 + 1.79e6i)10-s + 5.97e6i·11-s + (−1.11e7 − 1.74e7i)12-s + 2.93e7·13-s + (−3.30e6 − 1.12e7i)14-s − 6.28e7i·15-s + (1.12e8 − 2.43e8i)16-s + 5.58e8·17-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s − 0.577i·3-s + (0.842 − 0.539i)4-s + 0.637·5-s + (0.162 + 0.554i)6-s + 0.111i·7-s + (−0.656 + 0.754i)8-s − 0.333·9-s + (−0.611 + 0.179i)10-s + 0.306i·11-s + (−0.311 − 0.486i)12-s + 0.466·13-s + (−0.0313 − 0.106i)14-s − 0.368i·15-s + (0.418 − 0.908i)16-s + 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 + 0.842i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.539 + 0.842i$
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: \((2,\ 12,\ (\ :7),\ 0.539 + 0.842i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(1.11549 - 0.610135i\)
\(L(\frac12)\) \(\approx\) \(1.11549 - 0.610135i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (122. - 35.9i)T \)
3 \( 1 + 1.26e3iT \)
good5 \( 1 - 4.98e4T + 6.10e9T^{2} \)
7 \( 1 - 9.17e4iT - 6.78e11T^{2} \)
11 \( 1 - 5.97e6iT - 3.79e14T^{2} \)
13 \( 1 - 2.93e7T + 3.93e15T^{2} \)
17 \( 1 - 5.58e8T + 1.68e17T^{2} \)
19 \( 1 + 1.48e9iT - 7.99e17T^{2} \)
23 \( 1 + 2.13e9iT - 1.15e19T^{2} \)
29 \( 1 - 1.57e10T + 2.97e20T^{2} \)
31 \( 1 + 3.63e10iT - 7.56e20T^{2} \)
37 \( 1 + 3.90e10T + 9.01e21T^{2} \)
41 \( 1 - 5.94e10T + 3.79e22T^{2} \)
43 \( 1 + 3.53e11iT - 7.38e22T^{2} \)
47 \( 1 + 8.01e11iT - 2.56e23T^{2} \)
53 \( 1 - 1.45e12T + 1.37e24T^{2} \)
59 \( 1 - 4.73e12iT - 6.19e24T^{2} \)
61 \( 1 + 4.10e12T + 9.87e24T^{2} \)
67 \( 1 - 4.11e12iT - 3.67e25T^{2} \)
71 \( 1 - 9.30e12iT - 8.27e25T^{2} \)
73 \( 1 - 1.43e12T + 1.22e26T^{2} \)
79 \( 1 + 5.55e12iT - 3.68e26T^{2} \)
83 \( 1 + 1.50e13iT - 7.36e26T^{2} \)
89 \( 1 + 1.54e13T + 1.95e27T^{2} \)
97 \( 1 - 1.27e14T + 6.52e27T^{2} \)
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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

−16.79690579760951366682522885023, −15.22245091054312768742177814683, −13.67536795671318611151361548549, −11.88147260061012800975806923379, −10.23634572375278624835207297448, −8.781824067777144952145921620732, −7.20094735684242332396108480400, −5.75150474246273791507030944038, −2.36388626580380895000155630817, −0.78570172223900011479665779760, 1.36810460191124394755642699685, 3.36123023395864350153872346175, 5.94719344897090801354517023152, 8.034027283984947812213728414801, 9.588150172619655864895464349619, 10.58445446660065470207761040139, 12.19522929687524795653403932492, 14.17228316839115076800514260699, 15.92814388353153038249927551146, 16.92749748578880498399598213550

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