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Properties

Label 2-12-12.11-c13-0-14
Degree $2$
Conductor $12$
Sign $-0.303 + 0.952i$
Analytic cond. $12.8677$
Root an. cond. $3.58715$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.20 − 90.0i)2-s + (−618. + 1.10e3i)3-s + (−8.02e3 + 1.65e3i)4-s + 4.92e4i·5-s + (1.04e5 + 4.55e4i)6-s − 5.27e5i·7-s + (2.23e5 + 7.07e5i)8-s + (−8.28e5 − 1.36e6i)9-s + (4.43e6 − 4.52e5i)10-s + 4.09e6·11-s + (3.13e6 − 9.85e6i)12-s − 8.75e6·13-s + (−4.75e7 + 4.85e6i)14-s + (−5.41e7 − 3.04e7i)15-s + (6.16e7 − 2.65e7i)16-s − 5.78e7i·17-s + ⋯
L(s)  = 1  + (−0.101 − 0.994i)2-s + (−0.490 + 0.871i)3-s + (−0.979 + 0.202i)4-s + 1.40i·5-s + (0.917 + 0.398i)6-s − 1.69i·7-s + (0.300 + 0.953i)8-s + (−0.519 − 0.854i)9-s + (1.40 − 0.143i)10-s + 0.697·11-s + (0.303 − 0.952i)12-s − 0.503·13-s + (−1.68 + 0.172i)14-s + (−1.22 − 0.690i)15-s + (0.918 − 0.396i)16-s − 0.581i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.303 + 0.952i$
Analytic conductor: \(12.8677\)
Root analytic conductor: \(3.58715\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :13/2),\ -0.303 + 0.952i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.522732 - 0.715144i\)
\(L(\frac12)\) \(\approx\) \(0.522732 - 0.715144i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (9.20 + 90.0i)T \)
3 \( 1 + (618. - 1.10e3i)T \)
good5 \( 1 - 4.92e4iT - 1.22e9T^{2} \)
7 \( 1 + 5.27e5iT - 9.68e10T^{2} \)
11 \( 1 - 4.09e6T + 3.45e13T^{2} \)
13 \( 1 + 8.75e6T + 3.02e14T^{2} \)
17 \( 1 + 5.78e7iT - 9.90e15T^{2} \)
19 \( 1 + 2.75e8iT - 4.20e16T^{2} \)
23 \( 1 - 4.87e8T + 5.04e17T^{2} \)
29 \( 1 + 2.09e9iT - 1.02e19T^{2} \)
31 \( 1 + 7.35e8iT - 2.44e19T^{2} \)
37 \( 1 - 8.45e9T + 2.43e20T^{2} \)
41 \( 1 - 2.02e10iT - 9.25e20T^{2} \)
43 \( 1 + 6.24e10iT - 1.71e21T^{2} \)
47 \( 1 + 5.46e10T + 5.46e21T^{2} \)
53 \( 1 + 4.50e10iT - 2.60e22T^{2} \)
59 \( 1 + 1.17e11T + 1.04e23T^{2} \)
61 \( 1 + 1.79e11T + 1.61e23T^{2} \)
67 \( 1 - 2.45e11iT - 5.48e23T^{2} \)
71 \( 1 + 5.79e11T + 1.16e24T^{2} \)
73 \( 1 - 2.56e11T + 1.67e24T^{2} \)
79 \( 1 + 2.18e12iT - 4.66e24T^{2} \)
83 \( 1 - 1.53e12T + 8.87e24T^{2} \)
89 \( 1 - 2.56e12iT - 2.19e25T^{2} \)
97 \( 1 + 3.42e12T + 6.73e25T^{2} \)
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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

−16.90844426765369819266769572735, −14.81334092678725099116713956657, −13.69862917370570726752419527465, −11.45342024818355092224888961209, −10.67993288809757955186741734965, −9.635690526355179278846240045805, −7.00614111015253815417119090027, −4.43087819074739725696839702729, −3.13168980506165654887614665904, −0.46598538260412479500298104745, 1.38549858018686139405797343383, 5.06471363561114239920150038955, 6.11485494139621934635375518110, 8.139702183592885308791869004685, 9.155961695478277103433865895772, 12.15500035608411690798540004558, 12.83837159390919928865549957494, 14.66085711840652396118419848201, 16.24049997795332754179398813899, 17.14621420382190509634123495764

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