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Properties

Label 2-912-228.83-c1-0-28
Degree $2$
Conductor $912$
Sign $0.661 + 0.750i$
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  + (−1.63 + 0.578i)3-s + (3.06 − 1.77i)5-s − 0.587i·7-s + (2.33 − 1.88i)9-s + 3.00·11-s + (0.973 − 1.68i)13-s + (−3.98 + 4.66i)15-s + (−1.36 + 0.790i)17-s + (−4.34 + 0.308i)19-s + (0.339 + 0.958i)21-s + (1.35 − 2.34i)23-s + (3.77 − 6.53i)25-s + (−2.71 + 4.43i)27-s + (−4.59 − 2.65i)29-s − 4.43i·31-s + ⋯
L(s)  = 1  + (−0.942 + 0.334i)3-s + (1.37 − 0.792i)5-s − 0.221i·7-s + (0.776 − 0.629i)9-s + 0.907·11-s + (0.270 − 0.467i)13-s + (−1.02 + 1.20i)15-s + (−0.332 + 0.191i)17-s + (−0.997 + 0.0708i)19-s + (0.0741 + 0.209i)21-s + (0.282 − 0.489i)23-s + (0.755 − 1.30i)25-s + (−0.521 + 0.853i)27-s + (−0.853 − 0.492i)29-s − 0.796i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.661 + 0.750i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.661 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36744 - 0.617582i\)
\(L(\frac12)\) \(\approx\) \(1.36744 - 0.617582i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 - 0.578i)T \)
19 \( 1 + (4.34 - 0.308i)T \)
good5 \( 1 + (-3.06 + 1.77i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.587iT - 7T^{2} \)
11 \( 1 - 3.00T + 11T^{2} \)
13 \( 1 + (-0.973 + 1.68i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.36 - 0.790i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.35 + 2.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.59 + 2.65i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.43iT - 31T^{2} \)
37 \( 1 - 6.60T + 37T^{2} \)
41 \( 1 + (7.82 - 4.51i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.77 + 3.33i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.48 + 6.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.87 - 2.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.23 - 12.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0261 - 0.0452i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.662 - 0.382i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.86 + 10.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.10 + 7.10i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.80 + 4.50i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + (-9.20 - 5.31i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.27 + 9.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.06374981432198386061664182628, −9.231081947623991504225608702297, −8.642838095583336036722364687016, −7.20131790753437029930209091980, −6.11574566203348228897596286811, −5.85607306235441448771107767763, −4.72565174717648865833608717094, −3.95597920907342368681478845265, −2.09632308093831023530984691213, −0.881667555365772276509913209512, 1.47434735285282500159144770709, 2.41989158121715399048575913955, 3.98920725365769689052836782460, 5.22446344997549000318523586231, 6.03189830846442877591548567043, 6.62416244931708244944332844557, 7.24328740243148399629837901468, 8.739359205950761350775602435235, 9.504457908948121806267695970761, 10.29844984609416264293869150494

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