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Properties

Label 2-460-23.16-c1-0-5
Degree $2$
Conductor $460$
Sign $0.759 + 0.650i$
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  + (2.11 − 1.36i)3-s + (0.415 + 0.909i)5-s + (0.509 − 0.149i)7-s + (1.38 − 3.03i)9-s + (−3.18 − 3.67i)11-s + (5.61 + 1.64i)13-s + (2.11 + 1.36i)15-s + (0.910 − 6.33i)17-s + (0.565 + 3.92i)19-s + (0.874 − 1.00i)21-s + (−0.600 + 4.75i)23-s + (−0.654 + 0.755i)25-s + (−0.119 − 0.828i)27-s + (0.0305 − 0.212i)29-s + (−1.14 − 0.737i)31-s + ⋯
L(s)  = 1  + (1.22 − 0.785i)3-s + (0.185 + 0.406i)5-s + (0.192 − 0.0564i)7-s + (0.461 − 1.01i)9-s + (−0.960 − 1.10i)11-s + (1.55 + 0.457i)13-s + (0.546 + 0.351i)15-s + (0.220 − 1.53i)17-s + (0.129 + 0.901i)19-s + (0.190 − 0.220i)21-s + (−0.125 + 0.992i)23-s + (−0.130 + 0.151i)25-s + (−0.0229 − 0.159i)27-s + (0.00566 − 0.0394i)29-s + (−0.206 − 0.132i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.759 + 0.650i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01803 - 0.746393i\)
\(L(\frac12)\) \(\approx\) \(2.01803 - 0.746393i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (0.600 - 4.75i)T \)
good3 \( 1 + (-2.11 + 1.36i)T + (1.24 - 2.72i)T^{2} \)
7 \( 1 + (-0.509 + 0.149i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (3.18 + 3.67i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (-5.61 - 1.64i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.910 + 6.33i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.565 - 3.92i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-0.0305 + 0.212i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (1.14 + 0.737i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-1.15 + 2.52i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-0.0105 - 0.0231i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (9.51 - 6.11i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + 8.51T + 47T^{2} \)
53 \( 1 + (-7.70 + 2.26i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-6.75 - 1.98i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (6.51 + 4.18i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (7.29 - 8.42i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (2.48 - 2.86i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-1.02 - 7.12i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-3.25 - 0.955i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-2.96 + 6.49i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-13.1 + 8.47i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-0.237 - 0.520i)T + (-63.5 + 73.3i)T^{2} \)
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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

−11.08441496137957107944733182392, −9.915161589374920142132303127076, −8.937107546365865181040096005417, −8.156693798630055023193209165937, −7.54416831575875995215974610985, −6.44538936873638400503837813377, −5.40213209598127864797238297257, −3.59138968336859127381299493964, −2.86111855752749014837676006041, −1.48035435483145634433854909926, 1.91128066135253502188027772243, 3.21288463589518534439046600267, 4.23048130522121034939145568768, 5.20099343397808754638677107670, 6.52156905437374731984039568907, 8.035151427490701736256779856487, 8.399005141299575584186637527278, 9.275544754379502429622509429952, 10.27408581270479372119426478253, 10.70704718670387944431922820661

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