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Properties

Label 2-12-3.2-c12-0-2
Degree $2$
Conductor $12$
Sign $-0.612 + 0.790i$
Analytic cond. $10.9679$
Root an. cond. $3.31178$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−446. + 576. i)3-s + 1.16e4i·5-s − 2.42e4·7-s + (−1.32e5 − 5.14e5i)9-s − 1.35e6i·11-s − 5.79e6·13-s + (−6.70e6 − 5.19e6i)15-s − 4.63e7i·17-s − 3.36e7·19-s + (1.08e7 − 1.39e7i)21-s + 1.86e8i·23-s + 1.08e8·25-s + (3.55e8 + 1.53e8i)27-s + 2.36e8i·29-s − 5.29e7·31-s + ⋯
L(s)  = 1  + (−0.612 + 0.790i)3-s + 0.744i·5-s − 0.205·7-s + (−0.248 − 0.968i)9-s − 0.764i·11-s − 1.19·13-s + (−0.588 − 0.456i)15-s − 1.92i·17-s − 0.714·19-s + (0.126 − 0.162i)21-s + 1.25i·23-s + 0.445·25-s + (0.917 + 0.396i)27-s + 0.397i·29-s − 0.0596·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.612 + 0.790i$
Analytic conductor: \(10.9679\)
Root analytic conductor: \(3.31178\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :6),\ -0.612 + 0.790i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0360398 - 0.0735527i\)
\(L(\frac12)\) \(\approx\) \(0.0360398 - 0.0735527i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (446. - 576. i)T \)
good5 \( 1 - 1.16e4iT - 2.44e8T^{2} \)
7 \( 1 + 2.42e4T + 1.38e10T^{2} \)
11 \( 1 + 1.35e6iT - 3.13e12T^{2} \)
13 \( 1 + 5.79e6T + 2.32e13T^{2} \)
17 \( 1 + 4.63e7iT - 5.82e14T^{2} \)
19 \( 1 + 3.36e7T + 2.21e15T^{2} \)
23 \( 1 - 1.86e8iT - 2.19e16T^{2} \)
29 \( 1 - 2.36e8iT - 3.53e17T^{2} \)
31 \( 1 + 5.29e7T + 7.87e17T^{2} \)
37 \( 1 + 4.83e9T + 6.58e18T^{2} \)
41 \( 1 + 6.23e9iT - 2.25e19T^{2} \)
43 \( 1 + 6.78e9T + 3.99e19T^{2} \)
47 \( 1 - 1.19e10iT - 1.16e20T^{2} \)
53 \( 1 + 7.56e9iT - 4.91e20T^{2} \)
59 \( 1 - 5.06e10iT - 1.77e21T^{2} \)
61 \( 1 + 5.11e10T + 2.65e21T^{2} \)
67 \( 1 - 9.11e9T + 8.18e21T^{2} \)
71 \( 1 + 1.39e11iT - 1.64e22T^{2} \)
73 \( 1 + 6.80e9T + 2.29e22T^{2} \)
79 \( 1 - 3.21e11T + 5.90e22T^{2} \)
83 \( 1 - 5.38e11iT - 1.06e23T^{2} \)
89 \( 1 + 5.38e10iT - 2.46e23T^{2} \)
97 \( 1 - 7.08e10T + 6.93e23T^{2} \)
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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

−16.61761514551798187200638137704, −15.40095717411806657760948754093, −14.06809179829239557695710108263, −11.96262799757043770422694035451, −10.71870473312817361411899864878, −9.369141350748145982376216301225, −6.94587407128825939662437221548, −5.16268165712544154606823769145, −3.14022527767358556871365411968, −0.03841212460788293360784983071, 1.84112682147528561159047373807, 4.81689233841853419213670078827, 6.59854836794461351091220858985, 8.304855948832749257228341698240, 10.35642466465576030820290392247, 12.26587407539806324855247839460, 12.90816479779571620332290713898, 14.83454425717460248340589866306, 16.72013831505058351855484445601, 17.42937921634580025650189033668

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