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Properties

Label 2-912-228.11-c1-0-9
Degree $2$
Conductor $912$
Sign $0.844 - 0.535i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.631 − 1.61i)3-s + (1.81 + 1.05i)5-s + 3.85i·7-s + (−2.20 − 2.03i)9-s − 2.41·11-s + (2.57 + 4.45i)13-s + (2.84 − 2.27i)15-s + (2.36 + 1.36i)17-s + (−1.37 + 4.13i)19-s + (6.22 + 2.43i)21-s + (2.33 + 4.05i)23-s + (−0.293 − 0.508i)25-s + (−4.67 + 2.26i)27-s + (6.26 − 3.61i)29-s − 2.26i·31-s + ⋯
L(s)  = 1  + (0.364 − 0.931i)3-s + (0.813 + 0.469i)5-s + 1.45i·7-s + (−0.733 − 0.679i)9-s − 0.729·11-s + (0.713 + 1.23i)13-s + (0.734 − 0.586i)15-s + (0.573 + 0.330i)17-s + (−0.316 + 0.948i)19-s + (1.35 + 0.531i)21-s + (0.487 + 0.844i)23-s + (−0.0586 − 0.101i)25-s + (−0.900 + 0.435i)27-s + (1.16 − 0.671i)29-s − 0.406i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.844 - 0.535i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83515 + 0.532347i\)
\(L(\frac12)\) \(\approx\) \(1.83515 + 0.532347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.631 + 1.61i)T \)
19 \( 1 + (1.37 - 4.13i)T \)
good5 \( 1 + (-1.81 - 1.05i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.85iT - 7T^{2} \)
11 \( 1 + 2.41T + 11T^{2} \)
13 \( 1 + (-2.57 - 4.45i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.36 - 1.36i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.33 - 4.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.26 + 3.61i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.26iT - 31T^{2} \)
37 \( 1 + 4.73T + 37T^{2} \)
41 \( 1 + (-10.1 - 5.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.19 + 4.72i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.29 + 3.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.79 - 5.07i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.43 + 11.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.57 - 2.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.92 + 1.11i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.29 + 2.23i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.23 + 12.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.16 - 2.98i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + (-5.45 + 3.15i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.20 - 2.08i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.978666624413622539417004806204, −9.252009964112354668528392639194, −8.445465733400793853637285784062, −7.77858168298250804772773660574, −6.41652710483290014880959554031, −6.16957499693530239157763090042, −5.20182756969286050854953216662, −3.48299869268092379728809998869, −2.40164757873645051876531743030, −1.72841591694000678797496501180, 0.902794757383563436980618919917, 2.71004367897344253015547950919, 3.63773061413549599718847114449, 4.81637128650680742388697964169, 5.31891738743684823274819227193, 6.54789390874653773624229838341, 7.67837418347895016276004866584, 8.432563830968330833463000119342, 9.271499127860669206285411268566, 10.24923468745172781434992350273

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