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Properties

Label 2-12-4.3-c14-0-2
Degree $2$
Conductor $12$
Sign $-0.894 - 0.446i$
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  + (67.3 + 108. i)2-s − 1.26e3i·3-s + (−7.30e3 + 1.46e4i)4-s + 3.07e4·5-s + (1.37e5 − 8.50e4i)6-s + 3.68e5i·7-s + (−2.08e6 + 1.92e5i)8-s − 1.59e6·9-s + (2.07e6 + 3.34e6i)10-s + 3.40e7i·11-s + (1.85e7 + 9.23e6i)12-s − 3.66e7·13-s + (−4.01e7 + 2.48e7i)14-s − 3.88e7i·15-s + (−1.61e8 − 2.14e8i)16-s − 4.79e8·17-s + ⋯
L(s)  = 1  + (0.526 + 0.850i)2-s − 0.577i·3-s + (−0.446 + 0.894i)4-s + 0.393·5-s + (0.490 − 0.303i)6-s + 0.447i·7-s + (−0.995 + 0.0915i)8-s − 0.333·9-s + (0.207 + 0.334i)10-s + 1.74i·11-s + (0.516 + 0.257i)12-s − 0.584·13-s + (−0.380 + 0.235i)14-s − 0.227i·15-s + (−0.601 − 0.798i)16-s − 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.894 - 0.446i$
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: \((2,\ 12,\ (\ :7),\ -0.894 - 0.446i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.373649 + 1.58695i\)
\(L(\frac12)\) \(\approx\) \(0.373649 + 1.58695i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-67.3 - 108. i)T \)
3 \( 1 + 1.26e3iT \)
good5 \( 1 - 3.07e4T + 6.10e9T^{2} \)
7 \( 1 - 3.68e5iT - 6.78e11T^{2} \)
11 \( 1 - 3.40e7iT - 3.79e14T^{2} \)
13 \( 1 + 3.66e7T + 3.93e15T^{2} \)
17 \( 1 + 4.79e8T + 1.68e17T^{2} \)
19 \( 1 - 3.76e8iT - 7.99e17T^{2} \)
23 \( 1 - 1.93e9iT - 1.15e19T^{2} \)
29 \( 1 - 2.43e10T + 2.97e20T^{2} \)
31 \( 1 - 5.84e9iT - 7.56e20T^{2} \)
37 \( 1 - 1.15e11T + 9.01e21T^{2} \)
41 \( 1 - 2.02e11T + 3.79e22T^{2} \)
43 \( 1 - 3.29e11iT - 7.38e22T^{2} \)
47 \( 1 + 2.44e11iT - 2.56e23T^{2} \)
53 \( 1 + 1.47e12T + 1.37e24T^{2} \)
59 \( 1 + 2.31e12iT - 6.19e24T^{2} \)
61 \( 1 - 7.16e11T + 9.87e24T^{2} \)
67 \( 1 + 2.62e12iT - 3.67e25T^{2} \)
71 \( 1 + 1.80e13iT - 8.27e25T^{2} \)
73 \( 1 + 1.82e13T + 1.22e26T^{2} \)
79 \( 1 - 2.69e13iT - 3.68e26T^{2} \)
83 \( 1 - 1.90e13iT - 7.36e26T^{2} \)
89 \( 1 - 4.60e13T + 1.95e27T^{2} \)
97 \( 1 - 1.38e14T + 6.52e27T^{2} \)
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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

−17.40710199422995709991428496183, −15.63605603014307921445046318919, −14.49370658676822327111701331506, −13.07818872378891346102624389630, −12.04384730779624655254641011187, −9.459609869827289353169102922295, −7.69092711295853880481655761832, −6.35848245955166045097763146104, −4.68540818201860725594877862792, −2.28527134796179441735373767320, 0.53561527303860276226257884081, 2.71263290411705165689678988560, 4.34993742534498966759378701146, 6.00545964286926496589167047667, 8.893020369558903032106741685653, 10.37622152676448398747583117987, 11.45969804417610935008057606393, 13.31536190951793510290686781994, 14.28484328511934985776901969970, 15.89869650901578980530201224521

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