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Properties

Label 2-12-4.3-c20-0-13
Degree $2$
Conductor $12$
Sign $0.931 + 0.362i$
Analytic cond. $30.4216$
Root an. cond. $5.51558$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (578. + 845. i)2-s − 3.40e4i·3-s + (−3.80e5 + 9.77e5i)4-s − 1.01e7·5-s + (2.88e7 − 1.97e7i)6-s + 2.56e7i·7-s + (−1.04e9 + 2.43e8i)8-s − 1.16e9·9-s + (−5.85e9 − 8.56e9i)10-s + 2.46e9i·11-s + (3.33e10 + 1.29e10i)12-s + 1.51e11·13-s + (−2.16e10 + 1.48e10i)14-s + 3.45e11i·15-s + (−8.10e11 − 7.43e11i)16-s + 1.62e12·17-s + ⋯
L(s)  = 1  + (0.564 + 0.825i)2-s − 0.577i·3-s + (−0.362 + 0.931i)4-s − 1.03·5-s + (0.476 − 0.325i)6-s + 0.0906i·7-s + (−0.973 + 0.226i)8-s − 0.333·9-s + (−0.585 − 0.856i)10-s + 0.0949i·11-s + (0.538 + 0.209i)12-s + 1.10·13-s + (−0.0748 + 0.0511i)14-s + 0.599i·15-s + (−0.736 − 0.676i)16-s + 0.807·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.931 + 0.362i$
Analytic conductor: \(30.4216\)
Root analytic conductor: \(5.51558\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :10),\ 0.931 + 0.362i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(1.700753396\)
\(L(\frac12)\) \(\approx\) \(1.700753396\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-578. - 845. i)T \)
3 \( 1 + 3.40e4iT \)
good5 \( 1 + 1.01e7T + 9.53e13T^{2} \)
7 \( 1 - 2.56e7iT - 7.97e16T^{2} \)
11 \( 1 - 2.46e9iT - 6.72e20T^{2} \)
13 \( 1 - 1.51e11T + 1.90e22T^{2} \)
17 \( 1 - 1.62e12T + 4.06e24T^{2} \)
19 \( 1 + 1.08e13iT - 3.75e25T^{2} \)
23 \( 1 + 3.59e13iT - 1.71e27T^{2} \)
29 \( 1 - 5.31e14T + 1.76e29T^{2} \)
31 \( 1 + 2.12e13iT - 6.71e29T^{2} \)
37 \( 1 - 7.03e15T + 2.31e31T^{2} \)
41 \( 1 + 5.52e15T + 1.80e32T^{2} \)
43 \( 1 + 1.84e16iT - 4.67e32T^{2} \)
47 \( 1 - 5.35e16iT - 2.76e33T^{2} \)
53 \( 1 + 2.55e17T + 3.05e34T^{2} \)
59 \( 1 + 4.45e17iT - 2.61e35T^{2} \)
61 \( 1 + 4.92e17T + 5.08e35T^{2} \)
67 \( 1 - 3.83e17iT - 3.32e36T^{2} \)
71 \( 1 + 1.23e18iT - 1.05e37T^{2} \)
73 \( 1 + 3.02e18T + 1.84e37T^{2} \)
79 \( 1 + 1.74e19iT - 8.96e37T^{2} \)
83 \( 1 + 1.37e19iT - 2.40e38T^{2} \)
89 \( 1 - 1.36e18T + 9.72e38T^{2} \)
97 \( 1 + 1.27e20T + 5.43e39T^{2} \)
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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

−15.24558744781114995570801991016, −13.82116481011726538746208729538, −12.56020752540246554140182011193, −11.37905942232656870076497219259, −8.693238452353779123180805188505, −7.58252375611301534013681245137, −6.29282383482539272265739944242, −4.52987665405737229988119646874, −3.03352719514617739606477645898, −0.52877861386556083692043051270, 1.16027077532961986725904562521, 3.29931415891107508809756322911, 4.16636363151215456899929322257, 5.83606035583290585325805819456, 8.199638061547173360003116964168, 9.936738641655934408057162585810, 11.20937856848480827148293259675, 12.25379623810494238452909187397, 13.91699494173390997825728389771, 15.18839418021819486308292724639

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