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Properties

Label 2-12-12.11-c5-0-1
Degree $2$
Conductor $12$
Sign $0.310 - 0.950i$
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  + (−5.41 − 1.63i)2-s + (4.17 + 15.0i)3-s + (26.6 + 17.7i)4-s + 73.1i·5-s + (1.97 − 88.1i)6-s − 51.8i·7-s + (−115. − 139. i)8-s + (−208. + 125. i)9-s + (119. − 395. i)10-s + 371.·11-s + (−155. + 474. i)12-s + 424.·13-s + (−84.8 + 280. i)14-s + (−1.09e3 + 305. i)15-s + (394. + 944. i)16-s − 1.40e3i·17-s + ⋯
L(s)  = 1  + (−0.957 − 0.289i)2-s + (0.268 + 0.963i)3-s + (0.832 + 0.554i)4-s + 1.30i·5-s + (0.0223 − 0.999i)6-s − 0.399i·7-s + (−0.636 − 0.771i)8-s + (−0.856 + 0.516i)9-s + (0.378 − 1.25i)10-s + 0.925·11-s + (−0.310 + 0.950i)12-s + 0.696·13-s + (−0.115 + 0.382i)14-s + (−1.26 + 0.350i)15-s + (0.385 + 0.922i)16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.310 - 0.950i$
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: \((2,\ 12,\ (\ :5/2),\ 0.310 - 0.950i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.720480 + 0.522350i\)
\(L(\frac12)\) \(\approx\) \(0.720480 + 0.522350i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.41 + 1.63i)T \)
3 \( 1 + (-4.17 - 15.0i)T \)
good5 \( 1 - 73.1iT - 3.12e3T^{2} \)
7 \( 1 + 51.8iT - 1.68e4T^{2} \)
11 \( 1 - 371.T + 1.61e5T^{2} \)
13 \( 1 - 424.T + 3.71e5T^{2} \)
17 \( 1 + 1.40e3iT - 1.41e6T^{2} \)
19 \( 1 + 319. iT - 2.47e6T^{2} \)
23 \( 1 - 2.42e3T + 6.43e6T^{2} \)
29 \( 1 - 3.39e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.65e3iT - 2.86e7T^{2} \)
37 \( 1 - 395.T + 6.93e7T^{2} \)
41 \( 1 + 5.78e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.63e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.36e4T + 2.29e8T^{2} \)
53 \( 1 - 1.96e3iT - 4.18e8T^{2} \)
59 \( 1 + 7.86e3T + 7.14e8T^{2} \)
61 \( 1 + 9.36e3T + 8.44e8T^{2} \)
67 \( 1 + 3.78e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.03e4T + 1.80e9T^{2} \)
73 \( 1 + 6.20e4T + 2.07e9T^{2} \)
79 \( 1 - 6.42e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.00e4T + 3.93e9T^{2} \)
89 \( 1 + 5.66e4iT - 5.58e9T^{2} \)
97 \( 1 + 2.23e4T + 8.58e9T^{2} \)
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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

−19.45199699851742493719750683332, −18.19139557471574660024624047436, −16.78778705005953742185620635009, −15.48324930718777922317530584738, −14.15837973774647300467059230406, −11.39565255385928655604159961759, −10.45512048170442275697797484455, −9.026773222648669868502847748085, −6.95991806357111451970752064714, −3.26438044507686794124104050459, 1.30326050094125470645358086998, 6.15609171820708310619378131278, 8.182448829984718223718398639380, 9.151777804060623101682587669533, 11.63230671926544909808404051002, 12.98933195158476555523958966029, 14.88002608156964868447259065866, 16.58598960240632682480926104060, 17.51950640024087280510455324199, 18.94111375871197164782807148163

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