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Properties

Label 2-920-115.102-c1-0-28
Degree $2$
Conductor $920$
Sign $0.492 + 0.870i$
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  + (1.45 − 0.316i)3-s + (−1.46 − 1.69i)5-s + (3.74 − 2.04i)7-s + (−0.717 + 0.327i)9-s + (3.97 + 3.44i)11-s + (1.16 − 2.14i)13-s + (−2.66 − 1.99i)15-s + (−4.94 − 3.70i)17-s + (0.582 − 4.04i)19-s + (4.79 − 4.15i)21-s + (2.02 + 4.34i)23-s + (−0.712 + 4.94i)25-s + (−4.51 + 3.37i)27-s + (6.20 − 0.892i)29-s + (4.73 − 3.04i)31-s + ⋯
L(s)  = 1  + (0.838 − 0.182i)3-s + (−0.654 − 0.755i)5-s + (1.41 − 0.772i)7-s + (−0.239 + 0.109i)9-s + (1.19 + 1.03i)11-s + (0.324 − 0.594i)13-s + (−0.687 − 0.514i)15-s + (−1.20 − 0.898i)17-s + (0.133 − 0.928i)19-s + (1.04 − 0.906i)21-s + (0.422 + 0.906i)23-s + (−0.142 + 0.989i)25-s + (−0.867 + 0.649i)27-s + (1.15 − 0.165i)29-s + (0.850 − 0.546i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.492 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86495 - 1.08805i\)
\(L(\frac12)\) \(\approx\) \(1.86495 - 1.08805i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 + 1.69i)T \)
23 \( 1 + (-2.02 - 4.34i)T \)
good3 \( 1 + (-1.45 + 0.316i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (-3.74 + 2.04i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (-3.97 - 3.44i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-1.16 + 2.14i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (4.94 + 3.70i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.582 + 4.04i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-6.20 + 0.892i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-4.73 + 3.04i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (9.55 + 3.56i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (-0.166 + 0.365i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (1.81 + 8.33i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (0.536 + 0.536i)T + 47iT^{2} \)
53 \( 1 + (-1.45 - 2.66i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (0.860 + 2.93i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-4.49 - 7.00i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-7.51 - 0.537i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (-1.45 - 1.68i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.44 - 1.93i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (7.24 - 2.12i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (4.44 - 11.9i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (1.18 + 0.764i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (3.95 + 10.5i)T + (-73.3 + 63.5i)T^{2} \)
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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

−9.725723256563036984245022851923, −8.760253115858629774823997111008, −8.469643710513942791494923482536, −7.41026231235877329376091752683, −7.00802374401840335461051716629, −5.20084881540877778944585895694, −4.55419652957420779668204942780, −3.71200692364776199733403162843, −2.23160207653295482301321639481, −1.04926080614210165135263059413, 1.67389543075897161465080353400, 2.88171830294535592476850450854, 3.80846109810534385140024182041, 4.67104741427472500707757083595, 6.13266029523619471750435312331, 6.71039888024187838896461418505, 8.273471622642618347015571749145, 8.369780567110322026412529438797, 8.980866063092352121884131766137, 10.31990742759446842546043679027

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