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Properties

Label 2-920-115.64-c1-0-9
Degree $2$
Conductor $920$
Sign $0.993 + 0.112i$
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  + (−0.861 − 0.746i)3-s + (−1.32 + 1.80i)5-s + (−1.27 + 0.581i)7-s + (−0.242 − 1.68i)9-s + (1.21 − 0.357i)11-s + (−1.16 − 0.530i)13-s + (2.48 − 0.568i)15-s + (−1.84 − 2.87i)17-s + (4.80 + 3.08i)19-s + (1.53 + 0.449i)21-s + (3.37 + 3.40i)23-s + (−1.51 − 4.76i)25-s + (−2.89 + 4.50i)27-s + (7.61 − 4.89i)29-s + (3.69 + 4.25i)31-s + ⋯
L(s)  = 1  + (−0.497 − 0.430i)3-s + (−0.590 + 0.806i)5-s + (−0.481 + 0.219i)7-s + (−0.0807 − 0.561i)9-s + (0.367 − 0.107i)11-s + (−0.322 − 0.147i)13-s + (0.641 − 0.146i)15-s + (−0.447 − 0.696i)17-s + (1.10 + 0.708i)19-s + (0.334 + 0.0981i)21-s + (0.704 + 0.709i)23-s + (−0.302 − 0.953i)25-s + (−0.557 + 0.867i)27-s + (1.41 − 0.908i)29-s + (0.662 + 0.764i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.993 + 0.112i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06908 - 0.0603569i\)
\(L(\frac12)\) \(\approx\) \(1.06908 - 0.0603569i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 1.80i)T \)
23 \( 1 + (-3.37 - 3.40i)T \)
good3 \( 1 + (0.861 + 0.746i)T + (0.426 + 2.96i)T^{2} \)
7 \( 1 + (1.27 - 0.581i)T + (4.58 - 5.29i)T^{2} \)
11 \( 1 + (-1.21 + 0.357i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (1.16 + 0.530i)T + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (1.84 + 2.87i)T + (-7.06 + 15.4i)T^{2} \)
19 \( 1 + (-4.80 - 3.08i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-7.61 + 4.89i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-3.69 - 4.25i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-10.1 + 1.46i)T + (35.5 - 10.4i)T^{2} \)
41 \( 1 + (1.21 - 8.46i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-3.18 - 2.75i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + 4.46iT - 47T^{2} \)
53 \( 1 + (-9.61 + 4.39i)T + (34.7 - 40.0i)T^{2} \)
59 \( 1 + (-2.55 + 5.59i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (6.44 + 7.43i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-1.92 + 6.54i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (-12.9 - 3.79i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (2.65 - 4.12i)T + (-30.3 - 66.4i)T^{2} \)
79 \( 1 + (-2.28 + 5.00i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (12.2 - 1.76i)T + (79.6 - 23.3i)T^{2} \)
89 \( 1 + (-7.61 + 8.78i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-4.10 - 0.590i)T + (93.0 + 27.3i)T^{2} \)
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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

−9.944769271377265283952168274538, −9.453371395845408454444491655688, −8.226455828699203104236282503901, −7.36997197275830872069020313172, −6.60423465096516243395407626764, −6.03240231981111092833380530828, −4.79501582166852296123206612928, −3.52447701181907142236256085455, −2.75313229540785131564702014271, −0.843739204263995706111807304819, 0.851676809764015003327731057027, 2.67706276082585527188387194350, 4.08686961054849378014506972622, 4.70044131082935457872879766686, 5.57301784975051648139861182954, 6.69046187341195434987149768600, 7.57074818246096546382156081442, 8.524797809958273649516683776063, 9.254992984803321155005077582278, 10.14912457771201969864993198720

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