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Properties

Label 2-12-4.3-c4-0-2
Degree $2$
Conductor $12$
Sign $0.931 + 0.363i$
Analytic cond. $1.24043$
Root an. cond. $1.11375$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.30 − 2.25i)2-s + 5.19i·3-s + (5.81 − 14.9i)4-s − 22.8·5-s + (11.7 + 17.1i)6-s + 56.8i·7-s + (−14.4 − 62.3i)8-s − 27·9-s + (−75.4 + 51.5i)10-s − 134. i·11-s + (77.4 + 30.2i)12-s + 247.·13-s + (128. + 187. i)14-s − 118. i·15-s + (−188. − 173. i)16-s − 92.3·17-s + ⋯
L(s)  = 1  + (0.825 − 0.564i)2-s + 0.577i·3-s + (0.363 − 0.931i)4-s − 0.913·5-s + (0.325 + 0.476i)6-s + 1.16i·7-s + (−0.225 − 0.974i)8-s − 0.333·9-s + (−0.754 + 0.515i)10-s − 1.11i·11-s + (0.537 + 0.209i)12-s + 1.46·13-s + (0.654 + 0.958i)14-s − 0.527i·15-s + (−0.735 − 0.677i)16-s − 0.319·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.931 + 0.363i$
Analytic conductor: \(1.24043\)
Root analytic conductor: \(1.11375\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :2),\ 0.931 + 0.363i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.42252 - 0.267731i\)
\(L(\frac12)\) \(\approx\) \(1.42252 - 0.267731i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.30 + 2.25i)T \)
3 \( 1 - 5.19iT \)
good5 \( 1 + 22.8T + 625T^{2} \)
7 \( 1 - 56.8iT - 2.40e3T^{2} \)
11 \( 1 + 134. iT - 1.46e4T^{2} \)
13 \( 1 - 247.T + 2.85e4T^{2} \)
17 \( 1 + 92.3T + 8.35e4T^{2} \)
19 \( 1 - 29.5iT - 1.30e5T^{2} \)
23 \( 1 - 571. iT - 2.79e5T^{2} \)
29 \( 1 - 20.0T + 7.07e5T^{2} \)
31 \( 1 - 474. iT - 9.23e5T^{2} \)
37 \( 1 + 755.T + 1.87e6T^{2} \)
41 \( 1 - 541.T + 2.82e6T^{2} \)
43 \( 1 + 3.09e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.05e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.76e3T + 7.89e6T^{2} \)
59 \( 1 - 2.58e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.40e3T + 1.38e7T^{2} \)
67 \( 1 - 379. iT - 2.01e7T^{2} \)
71 \( 1 + 702. iT - 2.54e7T^{2} \)
73 \( 1 - 9.82e3T + 2.83e7T^{2} \)
79 \( 1 + 3.75e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.04e4iT - 4.74e7T^{2} \)
89 \( 1 + 1.19e4T + 6.27e7T^{2} \)
97 \( 1 + 2.19e3T + 8.85e7T^{2} \)
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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

−19.57795627576057764635671751419, −18.55426953650097154561085903522, −15.95113273373776447709724124032, −15.37684568829154557086285052757, −13.69443596423998722939989779762, −11.93045416105845690846734904875, −10.96109952932730072767385126388, −8.804601982129558388563576446641, −5.75172618691198996991299154693, −3.57829799171819653973573853193, 4.11501248978157856650536640493, 6.77211105201542335933981436986, 8.041765237760488827793366948790, 11.15466798320034113788122280773, 12.64736754533774355715483861521, 13.85046118940692215806213530802, 15.32837098804388346791714279967, 16.61504600799511890604454294588, 18.00714676075821619188813754103, 19.87480711758433571860369045386

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