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Properties

Label 2-12-12.11-c19-0-17
Degree $2$
Conductor $12$
Sign $0.302 + 0.952i$
Analytic cond. $27.4580$
Root an. cond. $5.24003$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−246. − 680. i)2-s + (1.28e4 − 3.15e4i)3-s + (−4.02e5 + 3.35e5i)4-s + 2.25e5i·5-s + (−2.46e7 − 9.68e5i)6-s + 9.84e7i·7-s + (3.27e8 + 1.91e8i)8-s + (−8.32e8 − 8.11e8i)9-s + (1.53e8 − 5.55e7i)10-s + 6.04e9·11-s + (5.41e9 + 1.70e10i)12-s + 4.74e10·13-s + (6.70e10 − 2.42e10i)14-s + (7.12e9 + 2.89e9i)15-s + (4.98e10 − 2.70e11i)16-s + 8.55e11i·17-s + ⋯
L(s)  = 1  + (−0.340 − 0.940i)2-s + (0.376 − 0.926i)3-s + (−0.768 + 0.639i)4-s + 0.0516i·5-s + (−0.999 − 0.0392i)6-s + 0.922i·7-s + (0.863 + 0.505i)8-s + (−0.716 − 0.698i)9-s + (0.0485 − 0.0175i)10-s + 0.773·11-s + (0.302 + 0.952i)12-s + 1.24·13-s + (0.867 − 0.313i)14-s + (0.0478 + 0.0194i)15-s + (0.181 − 0.983i)16-s + 1.74i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.952i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.302 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.302 + 0.952i$
Analytic conductor: \(27.4580\)
Root analytic conductor: \(5.24003\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :19/2),\ 0.302 + 0.952i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.844287021\)
\(L(\frac12)\) \(\approx\) \(1.844287021\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (246. + 680. i)T \)
3 \( 1 + (-1.28e4 + 3.15e4i)T \)
good5 \( 1 - 2.25e5iT - 1.90e13T^{2} \)
7 \( 1 - 9.84e7iT - 1.13e16T^{2} \)
11 \( 1 - 6.04e9T + 6.11e19T^{2} \)
13 \( 1 - 4.74e10T + 1.46e21T^{2} \)
17 \( 1 - 8.55e11iT - 2.39e23T^{2} \)
19 \( 1 + 3.94e11iT - 1.97e24T^{2} \)
23 \( 1 + 5.41e10T + 7.46e25T^{2} \)
29 \( 1 + 3.42e12iT - 6.10e27T^{2} \)
31 \( 1 + 2.38e14iT - 2.16e28T^{2} \)
37 \( 1 - 4.65e14T + 6.24e29T^{2} \)
41 \( 1 + 2.62e14iT - 4.39e30T^{2} \)
43 \( 1 - 2.87e13iT - 1.08e31T^{2} \)
47 \( 1 + 5.76e15T + 5.88e31T^{2} \)
53 \( 1 - 3.52e16iT - 5.77e32T^{2} \)
59 \( 1 - 1.05e17T + 4.42e33T^{2} \)
61 \( 1 + 5.09e16T + 8.34e33T^{2} \)
67 \( 1 + 1.29e17iT - 4.95e34T^{2} \)
71 \( 1 - 4.53e17T + 1.49e35T^{2} \)
73 \( 1 - 8.94e17T + 2.53e35T^{2} \)
79 \( 1 + 6.91e17iT - 1.13e36T^{2} \)
83 \( 1 - 2.08e18T + 2.90e36T^{2} \)
89 \( 1 + 2.97e18iT - 1.09e37T^{2} \)
97 \( 1 - 9.21e17T + 5.60e37T^{2} \)
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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

−14.86487374155000717722823793686, −13.35461183517321495834938142685, −12.33634553537274589819945151389, −11.11645722146823459492028849731, −9.116054750146157160720349170313, −8.195560449330679294050428180333, −6.16604612815841615831329739324, −3.68986762403361364378294355708, −2.19394943931718253813683280394, −1.02895650902107728308762832913, 0.893463380071435957067856761693, 3.66931746599993416237722226823, 4.99919222609535503262279963651, 6.83227617511172053752460626922, 8.467572633518650300208319971682, 9.638169773785422327463671924568, 10.96946581297834792666533487313, 13.64202048722460671818185589014, 14.47266092995944937340509424718, 15.99077889405484318075751336015

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