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Properties

Label 2-12-4.3-c8-0-7
Degree $2$
Conductor $12$
Sign $-0.826 + 0.562i$
Analytic cond. $4.88854$
Root an. cond. $2.21100$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.47 − 14.1i)2-s − 46.7i·3-s + (−144. − 211. i)4-s − 159.·5-s + (−661. − 349. i)6-s − 707. i·7-s + (−4.07e3 + 455. i)8-s − 2.18e3·9-s + (−1.19e3 + 2.25e3i)10-s − 1.68e4i·11-s + (−9.89e3 + 6.73e3i)12-s + 4.42e4·13-s + (−1.00e4 − 5.28e3i)14-s + 7.44e3i·15-s + (−2.39e4 + 6.09e4i)16-s + 1.29e5·17-s + ⋯
L(s)  = 1  + (0.467 − 0.884i)2-s − 0.577i·3-s + (−0.562 − 0.826i)4-s − 0.254·5-s + (−0.510 − 0.269i)6-s − 0.294i·7-s + (−0.993 + 0.111i)8-s − 0.333·9-s + (−0.119 + 0.225i)10-s − 1.14i·11-s + (−0.477 + 0.325i)12-s + 1.54·13-s + (−0.260 − 0.137i)14-s + 0.147i·15-s + (−0.366 + 0.930i)16-s + 1.54·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.826 + 0.562i$
Analytic conductor: \(4.88854\)
Root analytic conductor: \(2.21100\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :4),\ -0.826 + 0.562i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.484265 - 1.57119i\)
\(L(\frac12)\) \(\approx\) \(0.484265 - 1.57119i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.47 + 14.1i)T \)
3 \( 1 + 46.7iT \)
good5 \( 1 + 159.T + 3.90e5T^{2} \)
7 \( 1 + 707. iT - 5.76e6T^{2} \)
11 \( 1 + 1.68e4iT - 2.14e8T^{2} \)
13 \( 1 - 4.42e4T + 8.15e8T^{2} \)
17 \( 1 - 1.29e5T + 6.97e9T^{2} \)
19 \( 1 - 5.11e4iT - 1.69e10T^{2} \)
23 \( 1 + 3.91e5iT - 7.83e10T^{2} \)
29 \( 1 + 3.57e5T + 5.00e11T^{2} \)
31 \( 1 - 1.54e6iT - 8.52e11T^{2} \)
37 \( 1 - 1.98e6T + 3.51e12T^{2} \)
41 \( 1 + 2.28e6T + 7.98e12T^{2} \)
43 \( 1 - 1.84e6iT - 1.16e13T^{2} \)
47 \( 1 + 2.32e6iT - 2.38e13T^{2} \)
53 \( 1 - 2.67e6T + 6.22e13T^{2} \)
59 \( 1 - 9.97e5iT - 1.46e14T^{2} \)
61 \( 1 - 3.56e6T + 1.91e14T^{2} \)
67 \( 1 - 2.47e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.76e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.39e7T + 8.06e14T^{2} \)
79 \( 1 + 4.14e7iT - 1.51e15T^{2} \)
83 \( 1 - 2.43e7iT - 2.25e15T^{2} \)
89 \( 1 - 1.02e8T + 3.93e15T^{2} \)
97 \( 1 - 2.90e7T + 7.83e15T^{2} \)
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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

−18.32382289420788757316197048501, −16.34265367705092847749795001143, −14.36910461629669996269246140713, −13.31136095347439205508755783756, −11.88642852235407080962442893996, −10.58466093648040508299339547911, −8.459061227803366888650247861184, −5.94568766135316590565660757696, −3.48121479239966809363220508563, −1.01813774044010880746264053551, 3.83199219687741618398241615555, 5.70072299877603598419803455299, 7.77232414038990708589272811099, 9.504487318087688294821015104006, 11.75221551507289300917012286320, 13.40156496771002624974748182860, 14.96747535798093516233387829486, 15.81556498271180146084767460466, 17.14970930794032068975523683173, 18.52766745909009223990631270436

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