Properties

Label 2-912-57.8-c1-0-4
Degree $2$
Conductor $912$
Sign $0.0799 - 0.996i$
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  + (−1.68 + 0.396i)3-s + (−2.18 − 1.26i)5-s − 3.37·7-s + (2.68 − 1.33i)9-s − 3.46i·11-s + (2.87 − 1.65i)13-s + (4.18 + 1.26i)15-s + (−2.18 − 1.26i)17-s + (4 + 1.73i)19-s + (5.68 − 1.33i)21-s + (−7.93 + 4.57i)23-s + (0.686 + 1.18i)25-s + (−4 + 3.31i)27-s + (0.186 + 0.322i)29-s + 7.72i·31-s + ⋯
L(s)  = 1  + (−0.973 + 0.228i)3-s + (−0.977 − 0.564i)5-s − 1.27·7-s + (0.895 − 0.445i)9-s − 1.04i·11-s + (0.796 − 0.459i)13-s + (1.08 + 0.325i)15-s + (−0.530 − 0.306i)17-s + (0.917 + 0.397i)19-s + (1.24 − 0.291i)21-s + (−1.65 + 0.954i)23-s + (0.137 + 0.237i)25-s + (−0.769 + 0.638i)27-s + (0.0345 + 0.0598i)29-s + 1.38i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0799 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0799 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.0799 - 0.996i$
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: \((2,\ 912,\ (\ :1/2),\ 0.0799 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267506 + 0.246921i\)
\(L(\frac12)\) \(\approx\) \(0.267506 + 0.246921i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 0.396i)T \)
19 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 + (2.18 + 1.26i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (-2.87 + 1.65i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.18 + 1.26i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (7.93 - 4.57i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.186 - 0.322i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.72iT - 31T^{2} \)
37 \( 1 - 11.1iT - 37T^{2} \)
41 \( 1 + (3.18 - 5.51i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.87 + 10.1i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.813 + 0.469i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.81 - 4.87i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.813 + 1.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.24 + 0.718i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.18 + 10.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.87 - 4.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12.9 - 7.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (-0.813 - 1.40i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.44 - 1.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.32747021535565849773956507972, −9.588354093737792654839138669002, −8.613146235468520974989211396727, −7.78590423446157074680213915938, −6.67072296473410565744244921791, −5.97678808233427860600992962211, −5.12720929094183299820347678924, −3.86430467609551180482488048442, −3.34936401731513186087082592253, −0.966639648396388166099700232620, 0.25529920401353459718084040003, 2.23118072155728588046063924003, 3.76984374589751298527961054284, 4.32782704387718889544392684895, 5.79164439518417405738747662096, 6.50631149256716717840650498388, 7.18510710314316282867952367111, 7.912257579779243926819089219161, 9.290013793925504943104505828671, 9.991722745401895940244939648793

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