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Properties

Label 2-920-115.17-c1-0-2
Degree $2$
Conductor $920$
Sign $-0.592 - 0.805i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  + (−2.83 + 1.54i)3-s + (0.823 − 2.07i)5-s + (−2.50 − 3.34i)7-s + (4.02 − 6.26i)9-s + (−2.09 + 0.955i)11-s + (−3.94 − 2.95i)13-s + (0.884 + 7.17i)15-s + (0.306 + 4.28i)17-s + (3.62 + 4.18i)19-s + (12.2 + 5.61i)21-s + (3.71 − 3.03i)23-s + (−3.64 − 3.42i)25-s + (−1.02 + 14.3i)27-s + (4.00 + 3.46i)29-s + (−7.20 + 2.11i)31-s + ⋯
L(s)  = 1  + (−1.63 + 0.894i)3-s + (0.368 − 0.929i)5-s + (−0.946 − 1.26i)7-s + (1.34 − 2.08i)9-s + (−0.630 + 0.287i)11-s + (−1.09 − 0.819i)13-s + (0.228 + 1.85i)15-s + (0.0742 + 1.03i)17-s + (0.831 + 0.960i)19-s + (2.68 + 1.22i)21-s + (0.774 − 0.632i)23-s + (−0.728 − 0.684i)25-s + (−0.197 + 2.76i)27-s + (0.742 + 0.643i)29-s + (−1.29 + 0.379i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.592 - 0.805i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ -0.592 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0837829 + 0.165598i\)
\(L(\frac12)\) \(\approx\) \(0.0837829 + 0.165598i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.823 + 2.07i)T \)
23 \( 1 + (-3.71 + 3.03i)T \)
good3 \( 1 + (2.83 - 1.54i)T + (1.62 - 2.52i)T^{2} \)
7 \( 1 + (2.50 + 3.34i)T + (-1.97 + 6.71i)T^{2} \)
11 \( 1 + (2.09 - 0.955i)T + (7.20 - 8.31i)T^{2} \)
13 \( 1 + (3.94 + 2.95i)T + (3.66 + 12.4i)T^{2} \)
17 \( 1 + (-0.306 - 4.28i)T + (-16.8 + 2.41i)T^{2} \)
19 \( 1 + (-3.62 - 4.18i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-4.00 - 3.46i)T + (4.12 + 28.7i)T^{2} \)
31 \( 1 + (7.20 - 2.11i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (5.96 - 1.29i)T + (33.6 - 15.3i)T^{2} \)
41 \( 1 + (-2.67 + 1.72i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (0.506 + 0.928i)T + (-23.2 + 36.1i)T^{2} \)
47 \( 1 + (-1.89 - 1.89i)T + 47iT^{2} \)
53 \( 1 + (3.73 - 2.79i)T + (14.9 - 50.8i)T^{2} \)
59 \( 1 + (8.23 - 1.18i)T + (56.6 - 16.6i)T^{2} \)
61 \( 1 + (-3.86 - 13.1i)T + (-51.3 + 32.9i)T^{2} \)
67 \( 1 + (-0.861 - 2.31i)T + (-50.6 + 43.8i)T^{2} \)
71 \( 1 + (-0.667 + 1.46i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-7.05 - 0.504i)T + (72.2 + 10.3i)T^{2} \)
79 \( 1 + (1.31 + 9.15i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-0.975 - 4.48i)T + (-75.4 + 34.4i)T^{2} \)
89 \( 1 + (5.18 + 1.52i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (2.50 - 11.5i)T + (-88.2 - 40.2i)T^{2} \)
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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

−10.31042256379105599895903663057, −9.983215962227482101051983514311, −9.062741032275289503954161905944, −7.63711613868818060888937272858, −6.79206415574663885753570025206, −5.78683634163805859507763746453, −5.17447497566046315314832091582, −4.39313999368824206549147570860, −3.42186079914405525482868351923, −1.06720260261273801469105002148, 0.12655219061019602738754063666, 2.11871187376019358555380294640, 2.96230522334672703683708677720, 5.06451606490063737318388096889, 5.46306226348181802890759484853, 6.42890949017876676696019283091, 6.96919475692742993725002972037, 7.61902171234152373513131975248, 9.312963540121912796571625470545, 9.777380667618657758952631474994

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