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Properties

Label 2-12-4.3-c16-0-5
Degree $2$
Conductor $12$
Sign $0.720 - 0.693i$
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  + (−235. − 100. i)2-s − 3.78e3i·3-s + (4.54e4 + 4.72e4i)4-s + 6.99e5·5-s + (−3.79e5 + 8.92e5i)6-s + 4.77e6i·7-s + (−5.96e6 − 1.56e7i)8-s − 1.43e7·9-s + (−1.64e8 − 7.01e7i)10-s + 3.02e8i·11-s + (1.78e8 − 1.72e8i)12-s − 7.87e8·13-s + (4.78e8 − 1.12e9i)14-s − 2.64e9i·15-s + (−1.68e8 + 4.29e9i)16-s − 1.70e9·17-s + ⋯
L(s)  = 1  + (−0.920 − 0.391i)2-s − 0.577i·3-s + (0.693 + 0.720i)4-s + 1.78·5-s + (−0.226 + 0.531i)6-s + 0.828i·7-s + (−0.355 − 0.934i)8-s − 0.333·9-s + (−1.64 − 0.701i)10-s + 1.41i·11-s + (0.416 − 0.400i)12-s − 0.965·13-s + (0.324 − 0.762i)14-s − 1.03i·15-s + (−0.0392 + 0.999i)16-s − 0.244·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.720 - 0.693i$
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: \((2,\ 12,\ (\ :8),\ 0.720 - 0.693i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(1.28851 + 0.518977i\)
\(L(\frac12)\) \(\approx\) \(1.28851 + 0.518977i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (235. + 100. i)T \)
3 \( 1 + 3.78e3iT \)
good5 \( 1 - 6.99e5T + 1.52e11T^{2} \)
7 \( 1 - 4.77e6iT - 3.32e13T^{2} \)
11 \( 1 - 3.02e8iT - 4.59e16T^{2} \)
13 \( 1 + 7.87e8T + 6.65e17T^{2} \)
17 \( 1 + 1.70e9T + 4.86e19T^{2} \)
19 \( 1 - 1.50e10iT - 2.88e20T^{2} \)
23 \( 1 - 1.14e11iT - 6.13e21T^{2} \)
29 \( 1 + 5.20e11T + 2.50e23T^{2} \)
31 \( 1 + 3.94e11iT - 7.27e23T^{2} \)
37 \( 1 - 1.57e12T + 1.23e25T^{2} \)
41 \( 1 + 4.91e12T + 6.37e25T^{2} \)
43 \( 1 + 7.30e12iT - 1.36e26T^{2} \)
47 \( 1 - 2.40e13iT - 5.66e26T^{2} \)
53 \( 1 - 1.16e14T + 3.87e27T^{2} \)
59 \( 1 - 1.48e14iT - 2.15e28T^{2} \)
61 \( 1 - 1.03e14T + 3.67e28T^{2} \)
67 \( 1 + 3.79e14iT - 1.64e29T^{2} \)
71 \( 1 - 7.89e14iT - 4.16e29T^{2} \)
73 \( 1 - 1.80e14T + 6.50e29T^{2} \)
79 \( 1 + 1.58e15iT - 2.30e30T^{2} \)
83 \( 1 + 3.30e15iT - 5.07e30T^{2} \)
89 \( 1 + 1.42e14T + 1.54e31T^{2} \)
97 \( 1 + 4.90e14T + 6.14e31T^{2} \)
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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

−17.03665812656417108669312872889, −14.93731452203754935062962677015, −13.15147373921751585561086716889, −12.05158049865562713729547314881, −10.03904802070948610013459204584, −9.220738172453812633320128132883, −7.27295011234341637871557284371, −5.71692014255333647190394506328, −2.35516042657287899353635821793, −1.68597768775810568360603934721, 0.65933761585762836218873009392, 2.45264226776900564594363950765, 5.32205412930739887506246363199, 6.66719886837189343886673740511, 8.795342294258940737509002170900, 9.953852419284521693544545586486, 10.88669935862278269880732627615, 13.57087577959433564699150987356, 14.63733917299765003694683258598, 16.56003884894001639474946276048

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