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Properties

Label 2-12-4.3-c18-0-16
Degree $2$
Conductor $12$
Sign $-0.935 - 0.352i$
Analytic cond. $24.6463$
Root an. cond. $4.96450$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (421. − 291. i)2-s + 1.13e4i·3-s + (9.25e4 − 2.45e5i)4-s − 1.11e6·5-s + (3.30e6 + 4.78e6i)6-s − 6.89e6i·7-s + (−3.24e7 − 1.30e8i)8-s − 1.29e8·9-s + (−4.68e8 + 3.23e8i)10-s + 1.93e9i·11-s + (2.78e9 + 1.05e9i)12-s − 8.31e9·13-s + (−2.00e9 − 2.90e9i)14-s − 1.26e10i·15-s + (−5.16e10 − 4.53e10i)16-s − 1.99e11·17-s + ⋯
L(s)  = 1  + (0.822 − 0.568i)2-s + 0.577i·3-s + (0.352 − 0.935i)4-s − 0.569·5-s + (0.328 + 0.474i)6-s − 0.170i·7-s + (−0.241 − 0.970i)8-s − 0.333·9-s + (−0.468 + 0.323i)10-s + 0.818i·11-s + (0.540 + 0.203i)12-s − 0.783·13-s + (−0.0971 − 0.140i)14-s − 0.328i·15-s + (−0.750 − 0.660i)16-s − 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.935 - 0.352i$
Analytic conductor: \(24.6463\)
Root analytic conductor: \(4.96450\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :9),\ -0.935 - 0.352i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.2396870117\)
\(L(\frac12)\) \(\approx\) \(0.2396870117\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-421. + 291. i)T \)
3 \( 1 - 1.13e4iT \)
good5 \( 1 + 1.11e6T + 3.81e12T^{2} \)
7 \( 1 + 6.89e6iT - 1.62e15T^{2} \)
11 \( 1 - 1.93e9iT - 5.55e18T^{2} \)
13 \( 1 + 8.31e9T + 1.12e20T^{2} \)
17 \( 1 + 1.99e11T + 1.40e22T^{2} \)
19 \( 1 + 2.93e11iT - 1.04e23T^{2} \)
23 \( 1 - 1.31e12iT - 3.24e24T^{2} \)
29 \( 1 + 1.02e13T + 2.10e26T^{2} \)
31 \( 1 - 1.94e13iT - 6.99e26T^{2} \)
37 \( 1 + 1.32e14T + 1.68e28T^{2} \)
41 \( 1 - 3.30e14T + 1.07e29T^{2} \)
43 \( 1 - 1.42e14iT - 2.52e29T^{2} \)
47 \( 1 + 1.39e15iT - 1.25e30T^{2} \)
53 \( 1 - 2.59e15T + 1.08e31T^{2} \)
59 \( 1 + 6.79e15iT - 7.50e31T^{2} \)
61 \( 1 - 7.81e15T + 1.36e32T^{2} \)
67 \( 1 + 3.42e16iT - 7.40e32T^{2} \)
71 \( 1 + 4.99e15iT - 2.10e33T^{2} \)
73 \( 1 + 2.48e16T + 3.46e33T^{2} \)
79 \( 1 + 1.86e17iT - 1.43e34T^{2} \)
83 \( 1 - 3.53e17iT - 3.49e34T^{2} \)
89 \( 1 - 5.20e17T + 1.22e35T^{2} \)
97 \( 1 + 1.36e18T + 5.77e35T^{2} \)
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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

−14.98584607894859733594525414412, −13.43963393218855579207073514398, −11.97767663619576153586483207252, −10.80103251370106936096215165844, −9.364504344589690895089514864496, −7.02231044305507159608699178023, −5.01060095610657264381736917486, −3.91801001505942375245723887093, −2.23590490723625243708749590298, −0.05536921860702274642929630923, 2.41086293862143688155292511913, 4.10786857246404271868236201395, 5.86246165808913504735462650476, 7.27792776151996033414387252167, 8.563585484856246021468756911454, 11.26980792355435362127301337519, 12.43893944907280607265734460778, 13.68338361154741081483912118261, 14.98508507703710111389689460908, 16.23929734113677874356482658125

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