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Properties

Label 2-12-4.3-c8-0-0
Degree $2$
Conductor $12$
Sign $-0.279 - 0.960i$
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  + (−2.26 − 15.8i)2-s + 46.7i·3-s + (−245. + 71.6i)4-s − 538.·5-s + (740. − 105. i)6-s + 3.35e3i·7-s + (1.69e3 + 3.73e3i)8-s − 2.18e3·9-s + (1.21e3 + 8.52e3i)10-s − 8.48e3i·11-s + (−3.35e3 − 1.14e4i)12-s − 5.16e4·13-s + (5.30e4 − 7.58e3i)14-s − 2.51e4i·15-s + (5.52e4 − 3.52e4i)16-s + 1.91e3·17-s + ⋯
L(s)  = 1  + (−0.141 − 0.989i)2-s + 0.577i·3-s + (−0.960 + 0.279i)4-s − 0.860·5-s + (0.571 − 0.0816i)6-s + 1.39i·7-s + (0.412 + 0.910i)8-s − 0.333·9-s + (0.121 + 0.852i)10-s − 0.579i·11-s + (−0.161 − 0.554i)12-s − 1.80·13-s + (1.38 − 0.197i)14-s − 0.497i·15-s + (0.843 − 0.537i)16-s + 0.0229·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.279 - 0.960i$
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: \((2,\ 12,\ (\ :4),\ -0.279 - 0.960i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.280502 + 0.373995i\)
\(L(\frac12)\) \(\approx\) \(0.280502 + 0.373995i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.26 + 15.8i)T \)
3 \( 1 - 46.7iT \)
good5 \( 1 + 538.T + 3.90e5T^{2} \)
7 \( 1 - 3.35e3iT - 5.76e6T^{2} \)
11 \( 1 + 8.48e3iT - 2.14e8T^{2} \)
13 \( 1 + 5.16e4T + 8.15e8T^{2} \)
17 \( 1 - 1.91e3T + 6.97e9T^{2} \)
19 \( 1 + 3.39e4iT - 1.69e10T^{2} \)
23 \( 1 - 1.53e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.24e6T + 5.00e11T^{2} \)
31 \( 1 - 1.12e6iT - 8.52e11T^{2} \)
37 \( 1 - 9.78e5T + 3.51e12T^{2} \)
41 \( 1 + 2.49e6T + 7.98e12T^{2} \)
43 \( 1 - 5.36e6iT - 1.16e13T^{2} \)
47 \( 1 + 5.10e5iT - 2.38e13T^{2} \)
53 \( 1 + 1.06e7T + 6.22e13T^{2} \)
59 \( 1 + 1.74e7iT - 1.46e14T^{2} \)
61 \( 1 - 9.90e6T + 1.91e14T^{2} \)
67 \( 1 - 6.78e6iT - 4.06e14T^{2} \)
71 \( 1 - 3.84e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.09e7T + 8.06e14T^{2} \)
79 \( 1 + 4.48e6iT - 1.51e15T^{2} \)
83 \( 1 + 8.74e6iT - 2.25e15T^{2} \)
89 \( 1 + 5.29e7T + 3.93e15T^{2} \)
97 \( 1 - 9.42e6T + 7.83e15T^{2} \)
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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

−19.10230834173328426235278539820, −17.57281327307001294182293151506, −15.84826037554872150888378341799, −14.50770357624979038952964751186, −12.37213393896505672917014290982, −11.49858094338261369940197747116, −9.731295713375235350373472413322, −8.326518287020513038529983321334, −4.93111909104071004222790690958, −2.87567109500339431741378655810, 0.30604514313974497321553977596, 4.45404745542439324941034247368, 7.00298847129426112537570585286, 7.86240778925643494739309089255, 10.07341907004268275994458596268, 12.30152863694417667896517419476, 13.86079826648201114391950781475, 15.07118768971485212346950272573, 16.68190122854689981837569794396, 17.55682371661287334716889793902

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