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Properties

Label 2-12-4.3-c6-0-2
Degree $2$
Conductor $12$
Sign $0.671 - 0.741i$
Analytic cond. $2.76064$
Root an. cond. $1.66152$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.46 + 2.87i)2-s + 15.5i·3-s + (47.4 + 42.9i)4-s + 18.1·5-s + (−44.8 + 116. i)6-s − 321. i·7-s + (230. + 457. i)8-s − 243·9-s + (135. + 52.1i)10-s − 2.01e3i·11-s + (−669. + 739. i)12-s − 2.84e3·13-s + (924. − 2.39e3i)14-s + 282. i·15-s + (405. + 4.07e3i)16-s + 6.86e3·17-s + ⋯
L(s)  = 1  + (0.933 + 0.359i)2-s + 0.577i·3-s + (0.741 + 0.671i)4-s + 0.144·5-s + (−0.207 + 0.538i)6-s − 0.937i·7-s + (0.450 + 0.892i)8-s − 0.333·9-s + (0.135 + 0.0521i)10-s − 1.51i·11-s + (−0.387 + 0.427i)12-s − 1.29·13-s + (0.337 − 0.874i)14-s + 0.0836i·15-s + (0.0990 + 0.995i)16-s + 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.671 - 0.741i$
Analytic conductor: \(2.76064\)
Root analytic conductor: \(1.66152\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :3),\ 0.671 - 0.741i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.96687 + 0.872492i\)
\(L(\frac12)\) \(\approx\) \(1.96687 + 0.872492i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.46 - 2.87i)T \)
3 \( 1 - 15.5iT \)
good5 \( 1 - 18.1T + 1.56e4T^{2} \)
7 \( 1 + 321. iT - 1.17e5T^{2} \)
11 \( 1 + 2.01e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.84e3T + 4.82e6T^{2} \)
17 \( 1 - 6.86e3T + 2.41e7T^{2} \)
19 \( 1 - 9.83e3iT - 4.70e7T^{2} \)
23 \( 1 - 4.62e3iT - 1.48e8T^{2} \)
29 \( 1 + 1.77e3T + 5.94e8T^{2} \)
31 \( 1 + 1.62e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.22e4T + 2.56e9T^{2} \)
41 \( 1 + 1.46e3T + 4.75e9T^{2} \)
43 \( 1 - 4.12e4iT - 6.32e9T^{2} \)
47 \( 1 + 5.60e4iT - 1.07e10T^{2} \)
53 \( 1 - 6.51e4T + 2.21e10T^{2} \)
59 \( 1 - 2.41e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.04e5T + 5.15e10T^{2} \)
67 \( 1 - 8.37e4iT - 9.04e10T^{2} \)
71 \( 1 + 4.98e5iT - 1.28e11T^{2} \)
73 \( 1 - 4.95e4T + 1.51e11T^{2} \)
79 \( 1 + 8.96e5iT - 2.43e11T^{2} \)
83 \( 1 - 6.68e5iT - 3.26e11T^{2} \)
89 \( 1 - 2.19e5T + 4.96e11T^{2} \)
97 \( 1 - 1.76e5T + 8.32e11T^{2} \)
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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

−19.37856613498623495773329321689, −17.01473424127231338326069167278, −16.34679832915544498340652901556, −14.63719107779964660401951032110, −13.70598050959148834767969199254, −11.89946272249858519136939563429, −10.26685767160733833444495179792, −7.77929066554938547993152078376, −5.63815772342515592872505611310, −3.63164797324934466661251804311, 2.31456411752238870578690078242, 5.18725548164028441227712165290, 7.13911800758783397918168428277, 9.811099566533262993915216706015, 11.92822936485764427040221210233, 12.67880010916371757062566574082, 14.38060844447914146478589727071, 15.44240921408206359212344360237, 17.46304879474166458829957217286, 18.99023373347988796771123569303

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