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Properties

Label 2-920-115.102-c1-0-1
Degree $2$
Conductor $920$
Sign $-0.990 - 0.134i$
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  + (−3.02 + 0.658i)3-s + (−1.35 − 1.77i)5-s + (−1.76 + 0.963i)7-s + (5.99 − 2.73i)9-s + (4.05 + 3.51i)11-s + (1.60 − 2.94i)13-s + (5.28 + 4.48i)15-s + (1.18 + 0.885i)17-s + (0.835 − 5.80i)19-s + (4.70 − 4.07i)21-s + (−4.74 − 0.728i)23-s + (−1.30 + 4.82i)25-s + (−8.91 + 6.67i)27-s + (−1.72 + 0.247i)29-s + (−2.95 + 1.89i)31-s + ⋯
L(s)  = 1  + (−1.74 + 0.380i)3-s + (−0.607 − 0.794i)5-s + (−0.667 + 0.364i)7-s + (1.99 − 0.913i)9-s + (1.22 + 1.05i)11-s + (0.446 − 0.817i)13-s + (1.36 + 1.15i)15-s + (0.286 + 0.214i)17-s + (0.191 − 1.33i)19-s + (1.02 − 0.890i)21-s + (−0.988 − 0.151i)23-s + (−0.261 + 0.965i)25-s + (−1.71 + 1.28i)27-s + (−0.319 + 0.0459i)29-s + (−0.530 + 0.341i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.990 - 0.134i$
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: \((2,\ 920,\ (\ :1/2),\ -0.990 - 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00289353 + 0.0428433i\)
\(L(\frac12)\) \(\approx\) \(0.00289353 + 0.0428433i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 + 1.77i)T \)
23 \( 1 + (4.74 + 0.728i)T \)
good3 \( 1 + (3.02 - 0.658i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (1.76 - 0.963i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (-4.05 - 3.51i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-1.60 + 2.94i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (-1.18 - 0.885i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.835 + 5.80i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (1.72 - 0.247i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (2.95 - 1.89i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (4.45 + 1.65i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (1.98 - 4.33i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-1.97 - 9.06i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (-2.41 - 2.41i)T + 47iT^{2} \)
53 \( 1 + (5.66 + 10.3i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (2.46 + 8.39i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (3.21 + 4.99i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (11.2 + 0.801i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (-2.63 - 3.04i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-2.50 - 3.34i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (11.2 - 3.31i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-3.49 + 9.36i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (-1.80 - 1.15i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-4.26 - 11.4i)T + (-73.3 + 63.5i)T^{2} \)
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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

−10.59043443941906444272451047954, −9.640207329876189702865206243219, −9.161477559142661259734950877348, −7.83271915353446448985277012366, −6.77849700015092118979262462405, −6.14970747374347993137899660386, −5.18151785550113966197165233184, −4.50520287459117815093456503374, −3.57227253006951735718118442311, −1.32877434627145972479667118831, 0.02947296028487830061262778186, 1.48736334346856854472835966122, 3.58673376150270849144386262734, 4.14780706240772170979567752756, 5.75829266108821133640443762703, 6.15602939957412737872959078264, 6.92044189099503100950867033700, 7.59278100190775435887052154849, 8.899013129749104236071049645594, 10.08713955627505760640776071274

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