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Properties

Label 2-12-4.3-c10-0-3
Degree $2$
Conductor $12$
Sign $0.376 - 0.926i$
Analytic cond. $7.62428$
Root an. cond. $2.76121$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.13 + 31.4i)2-s − 140. i·3-s + (−948. − 385. i)4-s + 5.10e3·5-s + (4.40e3 + 861. i)6-s + 1.48e4i·7-s + (1.79e4 − 2.74e4i)8-s − 1.96e4·9-s + (−3.13e4 + 1.60e5i)10-s + 2.24e5i·11-s + (−5.40e4 + 1.33e5i)12-s + 5.31e5·13-s + (−4.66e5 − 9.11e4i)14-s − 7.16e5i·15-s + (7.51e5 + 7.31e5i)16-s + 7.52e5·17-s + ⋯
L(s)  = 1  + (−0.191 + 0.981i)2-s − 0.577i·3-s + (−0.926 − 0.376i)4-s + 1.63·5-s + (0.566 + 0.110i)6-s + 0.883i·7-s + (0.547 − 0.836i)8-s − 0.333·9-s + (−0.313 + 1.60i)10-s + 1.39i·11-s + (−0.217 + 0.534i)12-s + 1.43·13-s + (−0.867 − 0.169i)14-s − 0.943i·15-s + (0.716 + 0.697i)16-s + 0.529·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.376 - 0.926i$
Analytic conductor: \(7.62428\)
Root analytic conductor: \(2.76121\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :5),\ 0.376 - 0.926i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.48690 + 1.00063i\)
\(L(\frac12)\) \(\approx\) \(1.48690 + 1.00063i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.13 - 31.4i)T \)
3 \( 1 + 140. iT \)
good5 \( 1 - 5.10e3T + 9.76e6T^{2} \)
7 \( 1 - 1.48e4iT - 2.82e8T^{2} \)
11 \( 1 - 2.24e5iT - 2.59e10T^{2} \)
13 \( 1 - 5.31e5T + 1.37e11T^{2} \)
17 \( 1 - 7.52e5T + 2.01e12T^{2} \)
19 \( 1 - 1.91e5iT - 6.13e12T^{2} \)
23 \( 1 + 6.15e6iT - 4.14e13T^{2} \)
29 \( 1 + 9.65e6T + 4.20e14T^{2} \)
31 \( 1 + 5.87e6iT - 8.19e14T^{2} \)
37 \( 1 + 3.56e7T + 4.80e15T^{2} \)
41 \( 1 + 1.96e8T + 1.34e16T^{2} \)
43 \( 1 - 8.35e7iT - 2.16e16T^{2} \)
47 \( 1 + 3.14e8iT - 5.25e16T^{2} \)
53 \( 1 - 3.41e7T + 1.74e17T^{2} \)
59 \( 1 + 3.81e8iT - 5.11e17T^{2} \)
61 \( 1 - 4.25e8T + 7.13e17T^{2} \)
67 \( 1 + 3.43e8iT - 1.82e18T^{2} \)
71 \( 1 + 1.10e9iT - 3.25e18T^{2} \)
73 \( 1 + 8.97e8T + 4.29e18T^{2} \)
79 \( 1 - 4.83e9iT - 9.46e18T^{2} \)
83 \( 1 + 2.01e9iT - 1.55e19T^{2} \)
89 \( 1 + 5.23e9T + 3.11e19T^{2} \)
97 \( 1 + 1.02e10T + 7.37e19T^{2} \)
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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

−18.02811807122916358526022573467, −16.83932022862903186943719749585, −15.11152401577271379429453816767, −13.82571785475894044559681290409, −12.71391839958610163762184115260, −9.992176487501755488464872307097, −8.677932113891130089849330897541, −6.61821379989655497976343081915, −5.46592735916751138691697651598, −1.73060297302932469729551524657, 1.24824244686522097284930966811, 3.44246072556503534055589385562, 5.66831885182708027051237255811, 8.779803736681583851677344449290, 10.10780309633076564670296201106, 11.08673854791380876842851272861, 13.44414348895036110576578955960, 13.92006795799641395834942559377, 16.53246381062322850311297892763, 17.54200956648081309015858137180

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