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Properties

Label 2-12-12.11-c19-0-28
Degree $2$
Conductor $12$
Sign $0.990 + 0.140i$
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  + (706. + 159. i)2-s + (3.25e4 − 1.01e4i)3-s + (4.73e5 + 2.24e5i)4-s − 5.54e6i·5-s + (2.46e7 − 1.98e6i)6-s + 1.49e8i·7-s + (2.98e8 + 2.34e8i)8-s + (9.56e8 − 6.59e8i)9-s + (8.81e8 − 3.91e9i)10-s + 2.49e6·11-s + (1.76e10 + 2.50e9i)12-s + 4.88e10·13-s + (−2.38e10 + 1.05e11i)14-s + (−5.62e10 − 1.80e11i)15-s + (1.73e11 + 2.12e11i)16-s − 4.51e11i·17-s + ⋯
L(s)  = 1  + (0.975 + 0.219i)2-s + (0.954 − 0.297i)3-s + (0.903 + 0.428i)4-s − 1.26i·5-s + (0.996 − 0.0804i)6-s + 1.40i·7-s + (0.787 + 0.616i)8-s + (0.823 − 0.567i)9-s + (0.278 − 1.23i)10-s + 0.000319·11-s + (0.990 + 0.140i)12-s + 1.27·13-s + (−0.308 + 1.37i)14-s + (−0.377 − 1.21i)15-s + (0.632 + 0.774i)16-s − 0.923i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.990 + 0.140i$
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: \((2,\ 12,\ (\ :19/2),\ 0.990 + 0.140i)\)

Particular Values

\(L(10)\) \(\approx\) \(5.503003596\)
\(L(\frac12)\) \(\approx\) \(5.503003596\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-706. - 159. i)T \)
3 \( 1 + (-3.25e4 + 1.01e4i)T \)
good5 \( 1 + 5.54e6iT - 1.90e13T^{2} \)
7 \( 1 - 1.49e8iT - 1.13e16T^{2} \)
11 \( 1 - 2.49e6T + 6.11e19T^{2} \)
13 \( 1 - 4.88e10T + 1.46e21T^{2} \)
17 \( 1 + 4.51e11iT - 2.39e23T^{2} \)
19 \( 1 + 6.03e11iT - 1.97e24T^{2} \)
23 \( 1 + 7.40e12T + 7.46e25T^{2} \)
29 \( 1 + 1.10e14iT - 6.10e27T^{2} \)
31 \( 1 - 2.18e14iT - 2.16e28T^{2} \)
37 \( 1 - 4.99e14T + 6.24e29T^{2} \)
41 \( 1 + 1.26e13iT - 4.39e30T^{2} \)
43 \( 1 - 3.26e15iT - 1.08e31T^{2} \)
47 \( 1 + 1.12e16T + 5.88e31T^{2} \)
53 \( 1 - 3.23e16iT - 5.77e32T^{2} \)
59 \( 1 + 6.67e16T + 4.42e33T^{2} \)
61 \( 1 + 1.62e17T + 8.34e33T^{2} \)
67 \( 1 - 4.88e15iT - 4.95e34T^{2} \)
71 \( 1 - 2.98e17T + 1.49e35T^{2} \)
73 \( 1 + 9.07e17T + 2.53e35T^{2} \)
79 \( 1 - 3.92e17iT - 1.13e36T^{2} \)
83 \( 1 - 8.04e17T + 2.90e36T^{2} \)
89 \( 1 + 1.37e18iT - 1.09e37T^{2} \)
97 \( 1 + 1.68e18T + 5.60e37T^{2} \)
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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

−15.41298070422864152066354389572, −13.86429705118786140329488695579, −12.83298908913680154702984024068, −11.83337515000765848046008565368, −9.101924248145209085006346684386, −8.096543189769249071265556622478, −6.08065349148935565402041389789, −4.58857418840610483116148358365, −2.90484291594817233886761784775, −1.49875799043321831878255979999, 1.65119143720292475158381405866, 3.31172326824091601136866110502, 4.03679004378228702073912595242, 6.44455784716791539814448459300, 7.72073808504216075193834776636, 10.23138773735256273139299139752, 10.94856382353364257195850896557, 13.21715335566118332973835124330, 14.11199664131557877513379822791, 14.98029964034958058460064514538

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