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Properties

Label 2-460-115.107-c1-0-1
Degree $2$
Conductor $460$
Sign $0.722 - 0.691i$
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  + (−1.91 − 0.713i)3-s + (−2.19 − 0.428i)5-s + (−2.12 + 0.462i)7-s + (0.880 + 0.763i)9-s + (5.24 − 0.753i)11-s + (−0.463 + 2.12i)13-s + (3.89 + 2.38i)15-s + (−1.92 + 3.53i)17-s + (1.34 + 0.395i)19-s + (4.39 + 0.631i)21-s + (4.79 + 0.0606i)23-s + (4.63 + 1.87i)25-s + (1.79 + 3.28i)27-s + (2.00 + 6.82i)29-s + (1.08 − 2.36i)31-s + ⋯
L(s)  = 1  + (−1.10 − 0.411i)3-s + (−0.981 − 0.191i)5-s + (−0.802 + 0.174i)7-s + (0.293 + 0.254i)9-s + (1.58 − 0.227i)11-s + (−0.128 + 0.590i)13-s + (1.00 + 0.615i)15-s + (−0.468 + 0.857i)17-s + (0.309 + 0.0907i)19-s + (0.958 + 0.137i)21-s + (0.999 + 0.0126i)23-s + (0.926 + 0.375i)25-s + (0.345 + 0.632i)27-s + (0.372 + 1.26i)29-s + (0.194 − 0.425i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.722 - 0.691i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.568107 + 0.227896i\)
\(L(\frac12)\) \(\approx\) \(0.568107 + 0.227896i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 0.428i)T \)
23 \( 1 + (-4.79 - 0.0606i)T \)
good3 \( 1 + (1.91 + 0.713i)T + (2.26 + 1.96i)T^{2} \)
7 \( 1 + (2.12 - 0.462i)T + (6.36 - 2.90i)T^{2} \)
11 \( 1 + (-5.24 + 0.753i)T + (10.5 - 3.09i)T^{2} \)
13 \( 1 + (0.463 - 2.12i)T + (-11.8 - 5.40i)T^{2} \)
17 \( 1 + (1.92 - 3.53i)T + (-9.19 - 14.3i)T^{2} \)
19 \( 1 + (-1.34 - 0.395i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-2.00 - 6.82i)T + (-24.3 + 15.6i)T^{2} \)
31 \( 1 + (-1.08 + 2.36i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-0.339 - 4.74i)T + (-36.6 + 5.26i)T^{2} \)
41 \( 1 + (6.94 + 8.01i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-2.00 + 5.38i)T + (-32.4 - 28.1i)T^{2} \)
47 \( 1 + (-0.0375 - 0.0375i)T + 47iT^{2} \)
53 \( 1 + (-3.01 - 13.8i)T + (-48.2 + 22.0i)T^{2} \)
59 \( 1 + (-5.26 - 8.19i)T + (-24.5 + 53.6i)T^{2} \)
61 \( 1 + (-3.57 - 1.63i)T + (39.9 + 46.1i)T^{2} \)
67 \( 1 + (5.52 + 7.37i)T + (-18.8 + 64.2i)T^{2} \)
71 \( 1 + (2.01 - 14.0i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-5.73 + 3.12i)T + (39.4 - 61.4i)T^{2} \)
79 \( 1 + (-3.47 + 2.23i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (5.81 - 0.415i)T + (82.1 - 11.8i)T^{2} \)
89 \( 1 + (-1.93 - 4.23i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (11.0 + 0.788i)T + (96.0 + 13.8i)T^{2} \)
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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

−11.35742157769996309914618354609, −10.58067531515715802773945892750, −9.184832926554029972104526794020, −8.634875511195775196075796772988, −7.02574862522291318882659669534, −6.69656164378564549630845877365, −5.65088029992249848761991426035, −4.37480204330615575072034753763, −3.36712562096054632095362838346, −1.13712206470477167286774845063, 0.54233604801705811752661055529, 3.11392775491362837669652818060, 4.21201262655331952494829785057, 5.11390464892664409088012794565, 6.46115733642315299397288446119, 6.91425196880343757915605908295, 8.204102223723230361977230208200, 9.382101862715684983650787557448, 10.11781097420674214301625221285, 11.26609412471160227254455766007

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