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Properties

Label 2-1332-148.139-c0-0-1
Degree $2$
Conductor $1332$
Sign $0.815 + 0.578i$
Analytic cond. $0.664754$
Root an. cond. $0.815324$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (0.223 − 0.266i)5-s + (0.866 − 0.5i)8-s + (0.173 − 0.300i)10-s + (−0.592 − 1.62i)13-s + (0.766 − 0.642i)16-s + (−0.524 + 1.43i)17-s + (0.118 − 0.326i)20-s + (0.152 + 0.866i)25-s + (−0.866 − 1.5i)26-s + (−1.62 + 0.939i)29-s + (0.642 − 0.766i)32-s + (−0.266 + 1.50i)34-s + (0.939 − 0.342i)37-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (0.223 − 0.266i)5-s + (0.866 − 0.5i)8-s + (0.173 − 0.300i)10-s + (−0.592 − 1.62i)13-s + (0.766 − 0.642i)16-s + (−0.524 + 1.43i)17-s + (0.118 − 0.326i)20-s + (0.152 + 0.866i)25-s + (−0.866 − 1.5i)26-s + (−1.62 + 0.939i)29-s + (0.642 − 0.766i)32-s + (−0.266 + 1.50i)34-s + (0.939 − 0.342i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1332\)    =    \(2^{2} \cdot 3^{2} \cdot 37\)
Sign: $0.815 + 0.578i$
Analytic conductor: \(0.664754\)
Root analytic conductor: \(0.815324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1332} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1332,\ (\ :0),\ 0.815 + 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.999225199\)
\(L(\frac12)\) \(\approx\) \(1.999225199\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 \)
37 \( 1 + (-0.939 + 0.342i)T \)
good5 \( 1 + (-0.223 + 0.266i)T + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \)
19 \( 1 + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (-1.20 + 0.439i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.32 - 1.11i)T + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.233 + 0.642i)T + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.766 - 0.642i)T^{2} \)
89 \( 1 + (0.984 + 1.17i)T + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.904954013899411001754664175730, −9.024052553641248413471286048354, −7.85082301808911264453205433688, −7.31185058835136623526987538299, −6.04020769334551485298489728345, −5.61941314543059012488020025323, −4.66301244424118899099197730439, −3.68261977244478790353354280985, −2.74101537663210139117901299788, −1.52962721299531859093282589190, 2.01220736707903967828861421397, 2.77959075112200319137318246810, 4.11679521970029403666211294793, 4.68724811624372878095831495828, 5.72629960985783079780850624330, 6.62649344902658929859756363844, 7.15047965132912870438780036123, 8.032015971377623711747054819289, 9.261816434328694850023014371229, 9.803799589969373007668376549092

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