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Properties

Label 2-12-4.3-c18-0-13
Degree $2$
Conductor $12$
Sign $0.839 + 0.543i$
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  + (244. + 449. i)2-s + 1.13e4i·3-s + (−1.42e5 + 2.20e5i)4-s + 9.64e5·5-s + (−5.11e6 + 2.77e6i)6-s − 7.06e7i·7-s + (−1.33e8 − 1.03e7i)8-s − 1.29e8·9-s + (2.35e8 + 4.33e8i)10-s − 3.14e9i·11-s + (−2.50e9 − 1.61e9i)12-s − 4.14e9·13-s + (3.17e10 − 1.72e10i)14-s + 1.09e10i·15-s + (−2.80e10 − 6.27e10i)16-s − 4.74e10·17-s + ⋯
L(s)  = 1  + (0.477 + 0.878i)2-s + 0.577i·3-s + (−0.543 + 0.839i)4-s + 0.493·5-s + (−0.507 + 0.275i)6-s − 1.75i·7-s + (−0.997 − 0.0768i)8-s − 0.333·9-s + (0.235 + 0.433i)10-s − 1.33i·11-s + (−0.484 − 0.313i)12-s − 0.390·13-s + (1.53 − 0.836i)14-s + 0.285i·15-s + (−0.408 − 0.912i)16-s − 0.399·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.839 + 0.543i$
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: \((2,\ 12,\ (\ :9),\ 0.839 + 0.543i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(1.552500742\)
\(L(\frac12)\) \(\approx\) \(1.552500742\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-244. - 449. i)T \)
3 \( 1 - 1.13e4iT \)
good5 \( 1 - 9.64e5T + 3.81e12T^{2} \)
7 \( 1 + 7.06e7iT - 1.62e15T^{2} \)
11 \( 1 + 3.14e9iT - 5.55e18T^{2} \)
13 \( 1 + 4.14e9T + 1.12e20T^{2} \)
17 \( 1 + 4.74e10T + 1.40e22T^{2} \)
19 \( 1 + 3.01e11iT - 1.04e23T^{2} \)
23 \( 1 - 2.32e12iT - 3.24e24T^{2} \)
29 \( 1 - 2.19e13T + 2.10e26T^{2} \)
31 \( 1 + 3.74e13iT - 6.99e26T^{2} \)
37 \( 1 - 1.06e13T + 1.68e28T^{2} \)
41 \( 1 + 1.41e14T + 1.07e29T^{2} \)
43 \( 1 - 2.98e14iT - 2.52e29T^{2} \)
47 \( 1 - 5.09e14iT - 1.25e30T^{2} \)
53 \( 1 + 3.69e15T + 1.08e31T^{2} \)
59 \( 1 + 1.17e16iT - 7.50e31T^{2} \)
61 \( 1 + 1.94e16T + 1.36e32T^{2} \)
67 \( 1 + 7.96e15iT - 7.40e32T^{2} \)
71 \( 1 + 5.54e16iT - 2.10e33T^{2} \)
73 \( 1 - 9.01e16T + 3.46e33T^{2} \)
79 \( 1 - 3.13e16iT - 1.43e34T^{2} \)
83 \( 1 + 2.89e16iT - 3.49e34T^{2} \)
89 \( 1 - 1.31e17T + 1.22e35T^{2} \)
97 \( 1 + 1.00e17T + 5.77e35T^{2} \)
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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

−15.65082007387995901090851034332, −13.97987305362615262535402744152, −13.46414719227868170628041873972, −11.16290515207345224815160495230, −9.562369251538601600576081877662, −7.82738508572747264102795113781, −6.29754506426612755663894043767, −4.67236030660081875672874732183, −3.37120136527182517768828460541, −0.42959127827145353386102646589, 1.76150495995473529222149712338, 2.60144351861930577675358116993, 4.93587291626174346090976898782, 6.27898341387469755344071944828, 8.731192325399245032481267395710, 10.08646806915749841559987451028, 12.07830440012693865786868444136, 12.53415208310475977571730935713, 14.22398403976435508035144381080, 15.35540806229722742354542385418

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