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Properties

Label 2-912-57.8-c1-0-13
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: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.740 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59727 + 0.616266i\)
\(L(\frac12)\) \(\approx\) \(1.59727 + 0.616266i\)
\(L(\frac{3}{2})\) 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

−10.17614967966998823992201184713, −9.172428181255919580016186242889, −8.518617305837231037918757096323, −7.81281260489277853597789546328, −7.18237000577368599398309037919, −5.63511820002696833629185512855, −4.62241744799044566342074424595, −3.79783544588350474451481572318, −3.11917017383792559019208721377, −1.29462789201440182915391793452, 0.899380398092860495822688419576, 2.79498988831039814300431250555, 3.45632974897786380173707300590, 4.18931292424986274499932195753, 5.95936047713614048775425752704, 6.74944725788585291189898291808, 7.59478109039139328938568348473, 8.165936736842675631936736907458, 8.993033148778290452182234904626, 9.860045447105207803407856303584

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