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Properties

Label 2-12-4.3-c14-0-9
Degree $2$
Conductor $12$
Sign $0.400 + 0.916i$
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  + (26.2 + 125. i)2-s + 1.26e3i·3-s + (−1.50e4 + 6.56e3i)4-s − 4.03e4·5-s + (−1.58e5 + 3.30e4i)6-s − 4.21e5i·7-s + (−1.21e6 − 1.70e6i)8-s − 1.59e6·9-s + (−1.05e6 − 5.05e6i)10-s + 1.42e6i·11-s + (−8.29e6 − 1.89e7i)12-s − 2.76e7·13-s + (5.27e7 − 1.10e7i)14-s − 5.09e7i·15-s + (1.82e8 − 1.97e8i)16-s + 3.56e8·17-s + ⋯
L(s)  = 1  + (0.204 + 0.978i)2-s + 0.577i·3-s + (−0.916 + 0.400i)4-s − 0.516·5-s + (−0.565 + 0.118i)6-s − 0.511i·7-s + (−0.579 − 0.814i)8-s − 0.333·9-s + (−0.105 − 0.505i)10-s + 0.0730i·11-s + (−0.231 − 0.528i)12-s − 0.440·13-s + (0.500 − 0.104i)14-s − 0.298i·15-s + (0.678 − 0.734i)16-s + 0.869·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.400 + 0.916i$
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: \((2,\ 12,\ (\ :7),\ 0.400 + 0.916i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.237076 - 0.155060i\)
\(L(\frac12)\) \(\approx\) \(0.237076 - 0.155060i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-26.2 - 125. i)T \)
3 \( 1 - 1.26e3iT \)
good5 \( 1 + 4.03e4T + 6.10e9T^{2} \)
7 \( 1 + 4.21e5iT - 6.78e11T^{2} \)
11 \( 1 - 1.42e6iT - 3.79e14T^{2} \)
13 \( 1 + 2.76e7T + 3.93e15T^{2} \)
17 \( 1 - 3.56e8T + 1.68e17T^{2} \)
19 \( 1 + 3.85e8iT - 7.99e17T^{2} \)
23 \( 1 + 6.03e9iT - 1.15e19T^{2} \)
29 \( 1 + 1.53e10T + 2.97e20T^{2} \)
31 \( 1 - 3.72e10iT - 7.56e20T^{2} \)
37 \( 1 + 1.69e11T + 9.01e21T^{2} \)
41 \( 1 + 3.36e11T + 3.79e22T^{2} \)
43 \( 1 + 2.13e11iT - 7.38e22T^{2} \)
47 \( 1 + 4.76e11iT - 2.56e23T^{2} \)
53 \( 1 + 1.64e12T + 1.37e24T^{2} \)
59 \( 1 + 9.85e10iT - 6.19e24T^{2} \)
61 \( 1 - 2.21e12T + 9.87e24T^{2} \)
67 \( 1 - 9.25e12iT - 3.67e25T^{2} \)
71 \( 1 + 4.91e12iT - 8.27e25T^{2} \)
73 \( 1 - 8.74e12T + 1.22e26T^{2} \)
79 \( 1 - 5.31e12iT - 3.68e26T^{2} \)
83 \( 1 + 1.43e13iT - 7.36e26T^{2} \)
89 \( 1 - 1.13e13T + 1.95e27T^{2} \)
97 \( 1 + 8.88e13T + 6.52e27T^{2} \)
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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

−16.29477478796207104950966081556, −15.12238275879557536676420856448, −13.97162400303185740921288420897, −12.25003907623083319967241517338, −10.21439612473042673709401239695, −8.528694302200133656236478547976, −7.02079718892856582828974732004, −5.07277818730817403334009984411, −3.64486287061581745947291715005, −0.10729446479312400363926981633, 1.71572991322867274284609255670, 3.46843298766986930891094852079, 5.52149918128548123526339002437, 7.892426600815277929036813597723, 9.606539420062448703532770005093, 11.44717586725821263221950731315, 12.36167189875496425536426832502, 13.75555294264594146294013946515, 15.20816216075721727083902888738, 17.28373967819838260460741133163

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