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Properties

Label 2-912-76.71-c1-0-7
Degree $2$
Conductor $912$
Sign $0.857 - 0.513i$
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  + (−0.173 + 0.984i)3-s + (0.0469 + 0.0394i)5-s + (1.48 − 0.860i)7-s + (−0.939 − 0.342i)9-s + (0.740 + 0.427i)11-s + (0.588 − 0.103i)13-s + (−0.0469 + 0.0394i)15-s + (6.35 − 2.31i)17-s + (−3.13 + 3.02i)19-s + (0.588 + 1.61i)21-s + (3.05 + 3.64i)23-s + (−0.867 − 4.92i)25-s + (0.5 − 0.866i)27-s + (2.64 − 7.26i)29-s + (4.95 + 8.58i)31-s + ⋯
L(s)  = 1  + (−0.100 + 0.568i)3-s + (0.0210 + 0.0176i)5-s + (0.563 − 0.325i)7-s + (−0.313 − 0.114i)9-s + (0.223 + 0.128i)11-s + (0.163 − 0.0287i)13-s + (−0.0121 + 0.0101i)15-s + (1.54 − 0.561i)17-s + (−0.719 + 0.694i)19-s + (0.128 + 0.352i)21-s + (0.638 + 0.760i)23-s + (−0.173 − 0.984i)25-s + (0.0962 − 0.166i)27-s + (0.491 − 1.34i)29-s + (0.890 + 1.54i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.857 - 0.513i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.857 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63151 + 0.451142i\)
\(L(\frac12)\) \(\approx\) \(1.63151 + 0.451142i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (3.13 - 3.02i)T \)
good5 \( 1 + (-0.0469 - 0.0394i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-1.48 + 0.860i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.740 - 0.427i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.588 + 0.103i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-6.35 + 2.31i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-3.05 - 3.64i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.64 + 7.26i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.95 - 8.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.26iT - 37T^{2} \)
41 \( 1 + (-2.43 - 0.430i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.605 - 0.722i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-1.40 + 3.84i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-4.83 - 5.76i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.268 - 0.0978i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.13 - 2.63i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-10.1 - 3.70i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.45 - 1.22i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.65 + 9.39i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.354 - 2.01i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (10.0 - 5.77i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-10.3 + 1.83i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.48 - 4.06i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.17772097455206922134754468734, −9.514694932470071027678484060174, −8.395295274753017758889515382760, −7.82143159572581015602543743841, −6.69450468723595461938085507641, −5.72121815649738814636523258717, −4.80857511364949823667016812142, −3.94471648269378437157145674735, −2.82947526906145162424626551254, −1.19567756697530893877107556320, 1.06997527538100243579203997051, 2.33673039155900585335964922026, 3.57311828336233671163990063097, 4.85026807048781434407317022845, 5.71352745658302176190492611799, 6.59402882075682316108894093773, 7.52858495066525782615366417191, 8.320824128516416271790984278443, 9.013423714549237149042158376552, 10.06626081353025464772327348972

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