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Properties

Label 2-12-4.3-c18-0-17
Degree $2$
Conductor $12$
Sign $0.555 - 0.831i$
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  + (148. − 490. i)2-s − 1.13e4i·3-s + (−2.18e5 − 1.45e5i)4-s − 3.24e6·5-s + (−5.56e6 − 1.68e6i)6-s − 7.85e7i·7-s + (−1.03e8 + 8.52e7i)8-s − 1.29e8·9-s + (−4.82e8 + 1.59e9i)10-s − 2.80e8i·11-s + (−1.65e9 + 2.47e9i)12-s + 1.28e10·13-s + (−3.84e10 − 1.16e10i)14-s + 3.69e10i·15-s + (2.63e10 + 6.34e10i)16-s − 8.93e10·17-s + ⋯
L(s)  = 1  + (0.289 − 0.957i)2-s − 0.577i·3-s + (−0.831 − 0.555i)4-s − 1.66·5-s + (−0.552 − 0.167i)6-s − 1.94i·7-s + (−0.772 + 0.635i)8-s − 0.333·9-s + (−0.482 + 1.59i)10-s − 0.118i·11-s + (−0.320 + 0.480i)12-s + 1.20·13-s + (−1.86 − 0.564i)14-s + 0.960i·15-s + (0.383 + 0.923i)16-s − 0.753·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.555 - 0.831i$
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: \((2,\ 12,\ (\ :9),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.4865697565\)
\(L(\frac12)\) \(\approx\) \(0.4865697565\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-148. + 490. i)T \)
3 \( 1 + 1.13e4iT \)
good5 \( 1 + 3.24e6T + 3.81e12T^{2} \)
7 \( 1 + 7.85e7iT - 1.62e15T^{2} \)
11 \( 1 + 2.80e8iT - 5.55e18T^{2} \)
13 \( 1 - 1.28e10T + 1.12e20T^{2} \)
17 \( 1 + 8.93e10T + 1.40e22T^{2} \)
19 \( 1 + 2.10e11iT - 1.04e23T^{2} \)
23 \( 1 - 5.65e11iT - 3.24e24T^{2} \)
29 \( 1 - 8.21e12T + 2.10e26T^{2} \)
31 \( 1 - 4.08e12iT - 6.99e26T^{2} \)
37 \( 1 + 1.03e14T + 1.68e28T^{2} \)
41 \( 1 - 2.18e14T + 1.07e29T^{2} \)
43 \( 1 + 7.14e14iT - 2.52e29T^{2} \)
47 \( 1 + 1.15e14iT - 1.25e30T^{2} \)
53 \( 1 + 4.38e15T + 1.08e31T^{2} \)
59 \( 1 + 1.59e15iT - 7.50e31T^{2} \)
61 \( 1 - 9.13e15T + 1.36e32T^{2} \)
67 \( 1 - 1.99e16iT - 7.40e32T^{2} \)
71 \( 1 - 4.69e16iT - 2.10e33T^{2} \)
73 \( 1 + 8.14e16T + 3.46e33T^{2} \)
79 \( 1 - 5.80e16iT - 1.43e34T^{2} \)
83 \( 1 - 6.82e16iT - 3.49e34T^{2} \)
89 \( 1 + 4.94e17T + 1.22e35T^{2} \)
97 \( 1 - 1.02e18T + 5.77e35T^{2} \)
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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

−14.03183649435577168385280149826, −13.01923504802328715852743253444, −11.47422422273375561051165662049, −10.71993824728096611208868900398, −8.413152796615700422200075482082, −7.01478666581719188207133634772, −4.34033469865090122053279359750, −3.46332801797093957868633290238, −1.07944334112082106618069410089, −0.18609163160677499496479617198, 3.25013024859463096335173539785, 4.60402686455378297742628255821, 6.13971339136517832882208072168, 8.138440893875684444069863024011, 8.931213723206528969046577484091, 11.47935493916731493094902829995, 12.56041366862377713936738716379, 14.71750841178844495190517375907, 15.66487250222180526166088623647, 16.04504733100415445832922907458

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