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Properties

Label 2-12-12.11-c5-0-5
Degree $2$
Conductor $12$
Sign $0.551 + 0.834i$
Analytic cond. $1.92460$
Root an. cond. $1.38730$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.91 − 5.32i)2-s + (14.9 + 4.52i)3-s + (−24.6 − 20.4i)4-s − 35.2i·5-s + (52.7 − 70.6i)6-s + 190. i·7-s + (−155. + 91.8i)8-s + (202. + 134. i)9-s + (−187. − 67.7i)10-s − 33.3·11-s + (−275. − 416. i)12-s − 396.·13-s + (1.01e3 + 365. i)14-s + (159. − 526. i)15-s + (189. + 1.00e3i)16-s − 1.27e3i·17-s + ⋯
L(s)  = 1  + (0.339 − 0.940i)2-s + (0.956 + 0.290i)3-s + (−0.769 − 0.638i)4-s − 0.631i·5-s + (0.597 − 0.801i)6-s + 1.46i·7-s + (−0.861 + 0.507i)8-s + (0.831 + 0.555i)9-s + (−0.593 − 0.214i)10-s − 0.0830·11-s + (−0.551 − 0.834i)12-s − 0.650·13-s + (1.38 + 0.498i)14-s + (0.183 − 0.604i)15-s + (0.185 + 0.982i)16-s − 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.551 + 0.834i$
Analytic conductor: \(1.92460\)
Root analytic conductor: \(1.38730\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :5/2),\ 0.551 + 0.834i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.46393 - 0.787284i\)
\(L(\frac12)\) \(\approx\) \(1.46393 - 0.787284i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.91 + 5.32i)T \)
3 \( 1 + (-14.9 - 4.52i)T \)
good5 \( 1 + 35.2iT - 3.12e3T^{2} \)
7 \( 1 - 190. iT - 1.68e4T^{2} \)
11 \( 1 + 33.3T + 1.61e5T^{2} \)
13 \( 1 + 396.T + 3.71e5T^{2} \)
17 \( 1 + 1.27e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.33e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.23e3T + 6.43e6T^{2} \)
29 \( 1 + 3.98e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.67e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.21e3T + 6.93e7T^{2} \)
41 \( 1 - 9.04e3iT - 1.15e8T^{2} \)
43 \( 1 + 5.34e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.64e4T + 2.29e8T^{2} \)
53 \( 1 - 1.15e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.26e4T + 7.14e8T^{2} \)
61 \( 1 - 2.42e4T + 8.44e8T^{2} \)
67 \( 1 - 1.99e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.33e4T + 1.80e9T^{2} \)
73 \( 1 - 2.97e4T + 2.07e9T^{2} \)
79 \( 1 + 2.62e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.85e3T + 3.93e9T^{2} \)
89 \( 1 + 1.20e5iT - 5.58e9T^{2} \)
97 \( 1 + 7.56e4T + 8.58e9T^{2} \)
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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

−19.30745209438196175159431284681, −18.15058171968610137871379412445, −15.79567155113623207193801670378, −14.56893189560432219646993349059, −13.10322335121197638623156180224, −11.84984495492303610012609723540, −9.688648986819870235206172859569, −8.652398431836808972887432529098, −4.95691342704886059388678804355, −2.55273244054702814622203562385, 3.82697122851037828727746900248, 6.85291047060060154795200766231, 8.022932933086208939709107198468, 10.08549236524544608945045972880, 12.81476245343586641031770486336, 14.08669317701496089993530905561, 14.83497648215956189693130763935, 16.53641554475757085530869367410, 17.91036644023036504641382329325, 19.33075423468823901885826432262

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