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Properties

Label 2-12-4.3-c20-0-10
Degree $2$
Conductor $12$
Sign $0.282 + 0.959i$
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  + (−146. − 1.01e3i)2-s − 3.40e4i·3-s + (−1.00e6 + 2.96e5i)4-s + 1.07e7·5-s + (−3.45e7 + 4.98e6i)6-s − 1.91e8i·7-s + (4.47e8 + 9.75e8i)8-s − 1.16e9·9-s + (−1.57e9 − 1.09e10i)10-s + 4.63e10i·11-s + (1.01e10 + 3.42e10i)12-s + 2.40e11·13-s + (−1.93e11 + 2.79e10i)14-s − 3.68e11i·15-s + (9.23e11 − 5.96e11i)16-s + 2.93e12·17-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s − 0.577i·3-s + (−0.959 + 0.282i)4-s + 1.10·5-s + (−0.571 + 0.0825i)6-s − 0.677i·7-s + (0.417 + 0.908i)8-s − 0.333·9-s + (−0.157 − 1.09i)10-s + 1.78i·11-s + (0.163 + 0.553i)12-s + 1.74·13-s + (−0.670 + 0.0967i)14-s − 0.638i·15-s + (0.839 − 0.542i)16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.282 + 0.959i$
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: \((2,\ 12,\ (\ :10),\ 0.282 + 0.959i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(2.250862405\)
\(L(\frac12)\) \(\approx\) \(2.250862405\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (146. + 1.01e3i)T \)
3 \( 1 + 3.40e4iT \)
good5 \( 1 - 1.07e7T + 9.53e13T^{2} \)
7 \( 1 + 1.91e8iT - 7.97e16T^{2} \)
11 \( 1 - 4.63e10iT - 6.72e20T^{2} \)
13 \( 1 - 2.40e11T + 1.90e22T^{2} \)
17 \( 1 - 2.93e12T + 4.06e24T^{2} \)
19 \( 1 - 7.95e12iT - 3.75e25T^{2} \)
23 \( 1 + 3.00e12iT - 1.71e27T^{2} \)
29 \( 1 + 3.33e14T + 1.76e29T^{2} \)
31 \( 1 + 6.82e14iT - 6.71e29T^{2} \)
37 \( 1 + 1.53e15T + 2.31e31T^{2} \)
41 \( 1 - 1.72e16T + 1.80e32T^{2} \)
43 \( 1 + 2.59e16iT - 4.67e32T^{2} \)
47 \( 1 - 3.90e16iT - 2.76e33T^{2} \)
53 \( 1 + 7.34e16T + 3.05e34T^{2} \)
59 \( 1 - 1.38e17iT - 2.61e35T^{2} \)
61 \( 1 + 2.55e16T + 5.08e35T^{2} \)
67 \( 1 - 1.07e18iT - 3.32e36T^{2} \)
71 \( 1 + 3.16e18iT - 1.05e37T^{2} \)
73 \( 1 - 3.57e18T + 1.84e37T^{2} \)
79 \( 1 + 2.44e18iT - 8.96e37T^{2} \)
83 \( 1 - 6.12e18iT - 2.40e38T^{2} \)
89 \( 1 - 4.51e19T + 9.72e38T^{2} \)
97 \( 1 + 5.96e19T + 5.43e39T^{2} \)
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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

−14.34920825725304776210450771598, −13.33452550214841607289760892238, −12.27601368665596897153615546219, −10.53810395669881481027133830650, −9.526250673413304480913761605028, −7.72433065158093371911094642422, −5.76563398038418664704304694466, −3.83381413804987672684536979950, −1.96369982633318695332089585265, −1.15813702109437265106576879228, 0.970111660856366392386707343089, 3.37147300447107417363074885382, 5.48797517100296914229134338565, 6.10678235886359865909927029399, 8.446186907604476962884358198234, 9.355227563317393815500709693764, 10.92487652761312826555955668197, 13.34071737182419454655104701817, 14.20844894942130341681262034443, 15.75609095867058115039842689836

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