Properties

Label 2-912-57.8-c1-0-2
Degree $2$
Conductor $912$
Sign $-0.245 - 0.969i$
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  + (0.821 − 1.52i)3-s + (−2.16 − 1.24i)5-s − 3.30·7-s + (−1.64 − 2.50i)9-s + 5.60i·11-s + (0.0750 − 0.0433i)13-s + (−3.67 + 2.26i)15-s + (4.69 + 2.71i)17-s + (−2.61 − 3.48i)19-s + (−2.71 + 5.04i)21-s + (−2.94 + 1.70i)23-s + (0.612 + 1.06i)25-s + (−5.17 + 0.453i)27-s + (1.78 + 3.09i)29-s + 6.88i·31-s + ⋯
L(s)  = 1  + (0.474 − 0.880i)3-s + (−0.966 − 0.557i)5-s − 1.25·7-s + (−0.549 − 0.835i)9-s + 1.69i·11-s + (0.0208 − 0.0120i)13-s + (−0.949 + 0.585i)15-s + (1.13 + 0.657i)17-s + (−0.600 − 0.799i)19-s + (−0.593 + 1.10i)21-s + (−0.614 + 0.354i)23-s + (0.122 + 0.212i)25-s + (−0.996 + 0.0872i)27-s + (0.332 + 0.575i)29-s + 1.23i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.245 - 0.969i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.245 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169432 + 0.217664i\)
\(L(\frac12)\) \(\approx\) \(0.169432 + 0.217664i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.821 + 1.52i)T \)
19 \( 1 + (2.61 + 3.48i)T \)
good5 \( 1 + (2.16 + 1.24i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.30T + 7T^{2} \)
11 \( 1 - 5.60iT - 11T^{2} \)
13 \( 1 + (-0.0750 + 0.0433i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.69 - 2.71i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.94 - 1.70i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.78 - 3.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.88iT - 31T^{2} \)
37 \( 1 + 1.83iT - 37T^{2} \)
41 \( 1 + (1.35 - 2.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.02 - 6.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.16 - 4.71i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.741 + 1.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.91 + 11.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.28 + 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.13 + 4.11i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.20 - 9.02i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.19 - 7.26i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.1 + 6.42i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + (0.0767 + 0.132i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.41 - 1.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.02887827673366888362333626502, −9.524855628969761391804086169879, −8.463643887146919610829203279809, −7.84385770866229167473356181024, −6.95850699128733218575443078784, −6.38805732147201590716333953847, −4.97868156012460412562146329802, −3.87473920631839030125720805151, −2.98191551200899299662179121975, −1.56185139917476233542912499760, 0.12131512485039009937992629509, 2.79357502846393320223146999258, 3.47478875712466282957856026101, 4.03957903631941181882614687987, 5.56936345237386867892690985775, 6.28156908991025540646380429352, 7.51443308033205894985981412813, 8.243289664011277945325121823781, 8.987770546085434397819658137894, 10.06096343403088477293333379071

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