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Properties

Label 2-460-23.9-c1-0-5
Degree $2$
Conductor $460$
Sign $0.590 + 0.807i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (−0.249 − 0.287i)3-s + (0.142 + 0.989i)5-s + (0.948 − 2.07i)7-s + (0.406 − 2.82i)9-s + (−1.15 − 0.338i)11-s + (−0.598 − 1.31i)13-s + (0.249 − 0.287i)15-s + (2.71 + 1.74i)17-s + (3.45 − 2.21i)19-s + (−0.834 + 0.244i)21-s + (0.285 − 4.78i)23-s + (−0.959 + 0.281i)25-s + (−1.87 + 1.20i)27-s + (7.48 + 4.80i)29-s + (3.34 − 3.85i)31-s + ⋯
L(s)  = 1  + (−0.144 − 0.166i)3-s + (0.0636 + 0.442i)5-s + (0.358 − 0.784i)7-s + (0.135 − 0.941i)9-s + (−0.347 − 0.101i)11-s + (−0.166 − 0.363i)13-s + (0.0644 − 0.0743i)15-s + (0.659 + 0.423i)17-s + (0.791 − 0.508i)19-s + (−0.182 + 0.0534i)21-s + (0.0596 − 0.998i)23-s + (−0.191 + 0.0563i)25-s + (−0.361 + 0.232i)27-s + (1.38 + 0.892i)29-s + (0.600 − 0.693i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.590 + 0.807i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.590 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21670 - 0.617669i\)
\(L(\frac12)\) \(\approx\) \(1.21670 - 0.617669i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-0.285 + 4.78i)T \)
good3 \( 1 + (0.249 + 0.287i)T + (-0.426 + 2.96i)T^{2} \)
7 \( 1 + (-0.948 + 2.07i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (1.15 + 0.338i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (0.598 + 1.31i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-2.71 - 1.74i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-3.45 + 2.21i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-7.48 - 4.80i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-3.34 + 3.85i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-0.528 + 3.67i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.982 + 6.83i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (6.09 + 7.03i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 5.96T + 47T^{2} \)
53 \( 1 + (5.25 - 11.5i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-2.77 - 6.08i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (8.39 - 9.68i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-6.69 + 1.96i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (-7.66 + 2.24i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (3.23 - 2.08i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-3.99 - 8.74i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-0.0525 + 0.365i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-6.81 - 7.86i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-1.48 - 10.3i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.72331029014439299211522745613, −10.27010310589876700988400274281, −9.181121578716794395447847739997, −8.068364941557681572645666097835, −7.18627925207510417612119444935, −6.39206472407179085682378425976, −5.22632109125693620437681037091, −3.98724733443639050267503598757, −2.85400791419741965671112691185, −0.956988850756816685261629953841, 1.68624781036009272551634830239, 3.08744065430744563371883234096, 4.80745673198029298306034549173, 5.21034549602435459728908227630, 6.44001481024870845166780445463, 7.85035437080796561963725954053, 8.269639233309992285218250999284, 9.617153891903391410068837381837, 10.07650483401695760448726332267, 11.44766365346568395265652464998

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