Properties

Label 2-912-57.50-c1-0-28
Degree $2$
Conductor $912$
Sign $0.740 + 0.671i$
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  + (1.5 − 0.866i)3-s + (3 − 1.73i)5-s − 7-s + (1.5 − 2.59i)9-s + 3.46i·11-s + (4.5 + 2.59i)13-s + (3 − 5.19i)15-s + (−3 + 1.73i)17-s + (4 − 1.73i)19-s + (−1.5 + 0.866i)21-s + (3.5 − 6.06i)25-s − 5.19i·27-s + (−3 + 5.19i)29-s − 1.73i·31-s + (2.99 + 5.19i)33-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (1.34 − 0.774i)5-s − 0.377·7-s + (0.5 − 0.866i)9-s + 1.04i·11-s + (1.24 + 0.720i)13-s + (0.774 − 1.34i)15-s + (−0.727 + 0.420i)17-s + (0.917 − 0.397i)19-s + (−0.327 + 0.188i)21-s + (0.700 − 1.21i)25-s − 0.999i·27-s + (−0.557 + 0.964i)29-s − 0.311i·31-s + (0.522 + 0.904i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.740 + 0.671i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.740 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48473 - 0.958669i\)
\(L(\frac12)\) \(\approx\) \(2.48473 - 0.958669i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
19 \( 1 + (-4 + 1.73i)T \)
good5 \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3 - 1.73i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + (6 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 + 3.46i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 + 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.669219937949775089496725376886, −9.095254492352062399537752274407, −8.670108693975763968142596658673, −7.37647689978811065529997490085, −6.61508926834439222043164833043, −5.78190990133850197054414974308, −4.63733297448008114901755583472, −3.51084002464723050754903134445, −2.11295748302503760662776943399, −1.44884665951825694270658039340, 1.65629867996553385547471978989, 3.01652075159725567265311669795, 3.37667335065790969181971061924, 4.95657816502970387590339440363, 6.00499403050244054769062561264, 6.55343454635500180073030035466, 7.88193314646383327811318753797, 8.584785539635644445243430999637, 9.595050222532212835509216804470, 9.928233454302836245203350097863

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