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Properties

Label 2-12-12.11-c7-0-10
Degree $2$
Conductor $12$
Sign $-0.413 + 0.910i$
Analytic cond. $3.74862$
Root an. cond. $1.93613$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.03 − 6.80i)2-s + (−46.2 − 6.77i)3-s + (35.4 − 122. i)4-s − 426. i·5-s + (−464. + 253. i)6-s + 780. i·7-s + (−516. − 1.35e3i)8-s + (2.09e3 + 626. i)9-s + (−2.90e3 − 3.85e3i)10-s + 1.92e3·11-s + (−2.47e3 + 5.45e3i)12-s + 8.02e3·13-s + (5.31e3 + 7.05e3i)14-s + (−2.89e3 + 1.97e4i)15-s + (−1.38e4 − 8.71e3i)16-s + 8.16e3i·17-s + ⋯
L(s)  = 1  + (0.798 − 0.601i)2-s + (−0.989 − 0.144i)3-s + (0.276 − 0.960i)4-s − 1.52i·5-s + (−0.877 + 0.479i)6-s + 0.860i·7-s + (−0.356 − 0.934i)8-s + (0.958 + 0.286i)9-s + (−0.917 − 1.21i)10-s + 0.435·11-s + (−0.413 + 0.910i)12-s + 1.01·13-s + (0.517 + 0.687i)14-s + (−0.221 + 1.51i)15-s + (−0.846 − 0.531i)16-s + 0.403i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.413 + 0.910i$
Analytic conductor: \(3.74862\)
Root analytic conductor: \(1.93613\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :7/2),\ -0.413 + 0.910i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.880170 - 1.36564i\)
\(L(\frac12)\) \(\approx\) \(0.880170 - 1.36564i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.03 + 6.80i)T \)
3 \( 1 + (46.2 + 6.77i)T \)
good5 \( 1 + 426. iT - 7.81e4T^{2} \)
7 \( 1 - 780. iT - 8.23e5T^{2} \)
11 \( 1 - 1.92e3T + 1.94e7T^{2} \)
13 \( 1 - 8.02e3T + 6.27e7T^{2} \)
17 \( 1 - 8.16e3iT - 4.10e8T^{2} \)
19 \( 1 + 3.02e4iT - 8.93e8T^{2} \)
23 \( 1 - 2.58e4T + 3.40e9T^{2} \)
29 \( 1 - 1.09e5iT - 1.72e10T^{2} \)
31 \( 1 + 3.60e4iT - 2.75e10T^{2} \)
37 \( 1 - 3.18e5T + 9.49e10T^{2} \)
41 \( 1 - 6.40e4iT - 1.94e11T^{2} \)
43 \( 1 - 7.94e5iT - 2.71e11T^{2} \)
47 \( 1 - 6.08e4T + 5.06e11T^{2} \)
53 \( 1 - 1.15e6iT - 1.17e12T^{2} \)
59 \( 1 - 8.05e5T + 2.48e12T^{2} \)
61 \( 1 + 2.13e6T + 3.14e12T^{2} \)
67 \( 1 + 4.05e6iT - 6.06e12T^{2} \)
71 \( 1 + 4.95e6T + 9.09e12T^{2} \)
73 \( 1 - 1.26e6T + 1.10e13T^{2} \)
79 \( 1 - 6.28e6iT - 1.92e13T^{2} \)
83 \( 1 - 3.29e6T + 2.71e13T^{2} \)
89 \( 1 + 1.58e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.60e6T + 8.07e13T^{2} \)
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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

−18.23759292498201445051480808649, −16.60240422543580044504304838852, −15.47127339941497649741070856547, −13.22808845029833396510392797547, −12.37311619646717165093799093972, −11.20633676697858315113176533870, −9.142677250942541027569305725621, −5.99387887763525521322903254639, −4.66845148338536239992438860703, −1.17235647672031556261493035009, 3.82154877893615284321762027149, 6.13458049156249601218468601606, 7.25164007334784795019689063093, 10.53765553109070247421002934513, 11.65685178006790307031497545085, 13.54342445455977598374667076208, 14.81700396884636779893692045596, 16.17900103840106440714993523723, 17.41818494452358918118010422160, 18.61567038854360996782488327744

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