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Properties

Label 2-912-57.50-c1-0-6
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 + 2·7-s + (1.5 + 2.59i)9-s + 1.73i·11-s + (−3 − 1.73i)13-s − 6·15-s + (−6 + 3.46i)17-s + (−0.5 + 4.33i)19-s + (3 + 1.73i)21-s + (3.5 − 6.06i)25-s + 5.19i·27-s + (3 − 5.19i)29-s − 6.92i·31-s + (−1.49 + 2.59i)33-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (−1.34 + 0.774i)5-s + 0.755·7-s + (0.5 + 0.866i)9-s + 0.522i·11-s + (−0.832 − 0.480i)13-s − 1.54·15-s + (−1.45 + 0.840i)17-s + (−0.114 + 0.993i)19-s + (0.654 + 0.377i)21-s + (0.700 − 1.21i)25-s + 0.999i·27-s + (0.557 − 0.964i)29-s − 1.24i·31-s + (−0.261 + 0.452i)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

\(L(1)\) \(\approx\) \(0.465597 + 1.20676i\)
\(L(\frac12)\) \(\approx\) \(0.465597 + 1.20676i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 4.33i)T \)
good5 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (6 - 3.46i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 + 1.73i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 + 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12 + 6.92i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.19iT - 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.5 + 4.33i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.41840626379780311296312983428, −9.661481057239537562704736764342, −8.478492103003179336485196609584, −7.895916634175371834038339563348, −7.43641298284835239214515304451, −6.24775450661600840688894080162, −4.57649365984010908194674420953, −4.27127357023424301750208968688, −3.10978147213474573631133134553, −2.08578192370435271478686706925, 0.53681737565738816201639661095, 2.08914749475677923902413133137, 3.31695438989022365791063079056, 4.45426573245439654641296831217, 4.96679150611870146457670861512, 6.78563242013722514051446588945, 7.30992682572251170484029342221, 8.194285299573093888085316549828, 8.824550681718801704146997609027, 9.287835119513140027982092698119

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