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Properties

Label 2-12-4.3-c14-0-13
Degree $2$
Conductor $12$
Sign $-0.357 + 0.933i$
Analytic cond. $14.9194$
Root an. cond. $3.86257$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (125. + 23.2i)2-s − 1.26e3i·3-s + (1.52e4 + 5.86e3i)4-s − 9.22e4·5-s + (2.94e4 − 1.58e5i)6-s − 1.24e6i·7-s + (1.78e6 + 1.09e6i)8-s − 1.59e6·9-s + (−1.16e7 − 2.14e6i)10-s − 1.79e7i·11-s + (7.40e6 − 1.93e7i)12-s − 8.85e7·13-s + (2.90e7 − 1.57e8i)14-s + 1.16e8i·15-s + (1.99e8 + 1.79e8i)16-s + 2.74e8·17-s + ⋯
L(s)  = 1  + (0.983 + 0.181i)2-s − 0.577i·3-s + (0.933 + 0.357i)4-s − 1.18·5-s + (0.105 − 0.567i)6-s − 1.51i·7-s + (0.853 + 0.521i)8-s − 0.333·9-s + (−1.16 − 0.214i)10-s − 0.921i·11-s + (0.206 − 0.539i)12-s − 1.41·13-s + (0.275 − 1.49i)14-s + 0.681i·15-s + (0.744 + 0.668i)16-s + 0.667·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.357 + 0.933i$
Analytic conductor: \(14.9194\)
Root analytic conductor: \(3.86257\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :7),\ -0.357 + 0.933i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(1.21818 - 1.77122i\)
\(L(\frac12)\) \(\approx\) \(1.21818 - 1.77122i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-125. - 23.2i)T \)
3 \( 1 + 1.26e3iT \)
good5 \( 1 + 9.22e4T + 6.10e9T^{2} \)
7 \( 1 + 1.24e6iT - 6.78e11T^{2} \)
11 \( 1 + 1.79e7iT - 3.79e14T^{2} \)
13 \( 1 + 8.85e7T + 3.93e15T^{2} \)
17 \( 1 - 2.74e8T + 1.68e17T^{2} \)
19 \( 1 + 1.46e9iT - 7.99e17T^{2} \)
23 \( 1 + 3.18e8iT - 1.15e19T^{2} \)
29 \( 1 - 7.35e9T + 2.97e20T^{2} \)
31 \( 1 - 4.19e10iT - 7.56e20T^{2} \)
37 \( 1 - 8.88e10T + 9.01e21T^{2} \)
41 \( 1 - 3.04e10T + 3.79e22T^{2} \)
43 \( 1 - 1.51e11iT - 7.38e22T^{2} \)
47 \( 1 - 8.00e10iT - 2.56e23T^{2} \)
53 \( 1 - 1.77e12T + 1.37e24T^{2} \)
59 \( 1 + 3.34e12iT - 6.19e24T^{2} \)
61 \( 1 + 2.99e12T + 9.87e24T^{2} \)
67 \( 1 + 6.69e12iT - 3.67e25T^{2} \)
71 \( 1 - 1.29e11iT - 8.27e25T^{2} \)
73 \( 1 - 2.97e11T + 1.22e26T^{2} \)
79 \( 1 + 1.67e13iT - 3.68e26T^{2} \)
83 \( 1 + 3.35e13iT - 7.36e26T^{2} \)
89 \( 1 - 6.37e13T + 1.95e27T^{2} \)
97 \( 1 + 7.48e13T + 6.52e27T^{2} \)
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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

−16.11698527262906109157582554633, −14.57139761282144526043436355948, −13.43693053455280697576786447233, −12.05287756599681716524883391381, −10.86844093105290742783439008556, −7.81029305535364356747297376801, −6.93522004572526283563489858333, −4.65300866863447908701082976742, −3.16147880790139276784026160665, −0.61872308499268369511177399400, 2.46832089886442467792302793943, 4.12881280856718924334576080025, 5.54955343905296087976866862308, 7.71907263883069994009763032591, 9.894290375895606954262690075139, 11.85421422433648385631361530002, 12.30363338392080855878706864444, 14.79300881176474731031228030624, 15.21027631599389098239073402037, 16.51772878181124416232704115327

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