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Properties

Label 2-12-4.3-c16-0-6
Degree $2$
Conductor $12$
Sign $0.323 - 0.946i$
Analytic cond. $19.4789$
Root an. cond. $4.41349$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (252. + 41.9i)2-s − 3.78e3i·3-s + (6.20e4 + 2.11e4i)4-s − 3.80e5·5-s + (1.58e5 − 9.56e5i)6-s + 8.07e6i·7-s + (1.47e7 + 7.94e6i)8-s − 1.43e7·9-s + (−9.61e7 − 1.59e7i)10-s + 2.42e8i·11-s + (8.02e7 − 2.34e8i)12-s + 1.45e9·13-s + (−3.38e8 + 2.03e9i)14-s + 1.44e9i·15-s + (3.39e9 + 2.62e9i)16-s − 2.24e9·17-s + ⋯
L(s)  = 1  + (0.986 + 0.163i)2-s − 0.577i·3-s + (0.946 + 0.323i)4-s − 0.974·5-s + (0.0945 − 0.569i)6-s + 1.40i·7-s + (0.880 + 0.473i)8-s − 0.333·9-s + (−0.961 − 0.159i)10-s + 1.12i·11-s + (0.186 − 0.546i)12-s + 1.78·13-s + (−0.229 + 1.38i)14-s + 0.562i·15-s + (0.791 + 0.611i)16-s − 0.322·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.323 - 0.946i$
Analytic conductor: \(19.4789\)
Root analytic conductor: \(4.41349\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :8),\ 0.323 - 0.946i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(2.36155 + 1.68903i\)
\(L(\frac12)\) \(\approx\) \(2.36155 + 1.68903i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-252. - 41.9i)T \)
3 \( 1 + 3.78e3iT \)
good5 \( 1 + 3.80e5T + 1.52e11T^{2} \)
7 \( 1 - 8.07e6iT - 3.32e13T^{2} \)
11 \( 1 - 2.42e8iT - 4.59e16T^{2} \)
13 \( 1 - 1.45e9T + 6.65e17T^{2} \)
17 \( 1 + 2.24e9T + 4.86e19T^{2} \)
19 \( 1 + 1.14e10iT - 2.88e20T^{2} \)
23 \( 1 - 1.29e11iT - 6.13e21T^{2} \)
29 \( 1 + 1.75e11T + 2.50e23T^{2} \)
31 \( 1 - 7.71e11iT - 7.27e23T^{2} \)
37 \( 1 + 2.05e12T + 1.23e25T^{2} \)
41 \( 1 + 1.95e12T + 6.37e25T^{2} \)
43 \( 1 + 1.76e13iT - 1.36e26T^{2} \)
47 \( 1 + 3.06e13iT - 5.66e26T^{2} \)
53 \( 1 - 1.13e12T + 3.87e27T^{2} \)
59 \( 1 + 4.12e12iT - 2.15e28T^{2} \)
61 \( 1 - 3.30e14T + 3.67e28T^{2} \)
67 \( 1 - 3.23e13iT - 1.64e29T^{2} \)
71 \( 1 - 2.47e14iT - 4.16e29T^{2} \)
73 \( 1 - 3.30e14T + 6.50e29T^{2} \)
79 \( 1 + 1.08e15iT - 2.30e30T^{2} \)
83 \( 1 - 2.59e15iT - 5.07e30T^{2} \)
89 \( 1 - 4.71e15T + 1.54e31T^{2} \)
97 \( 1 - 1.18e15T + 6.14e31T^{2} \)
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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

−15.74058336951088360014060384453, −15.26651141329875060025590087801, −13.45446251522264455464550593097, −12.20500175908552311908766854119, −11.36101687887849101172215644112, −8.530923997013644235070277349550, −7.01457942569317440059999557278, −5.46146961506698443442114233756, −3.60828806177371311057959225589, −1.89773771260177539758327761056, 0.801009426968903661628570182694, 3.50255142456106220743336343317, 4.21704240216928422223943615873, 6.26963478803777795756975663038, 8.087623044657729141024337677235, 10.63393858193231353130433332962, 11.34605464564157612431549959131, 13.24388691665173556999703754254, 14.35208217637176406778116934667, 15.90024220656130602886477302813

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