Properties

Label 2-460-115.103-c1-0-8
Degree $2$
Conductor $460$
Sign $-0.814 + 0.580i$
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  + (−1.93 + 1.44i)3-s + (1.00 − 1.99i)5-s + (0.0873 − 1.22i)7-s + (0.799 − 2.72i)9-s + (−2.76 + 4.30i)11-s + (−6.94 + 0.497i)13-s + (0.960 + 5.31i)15-s + (1.90 − 5.11i)17-s + (−1.12 + 2.45i)19-s + (1.60 + 2.48i)21-s + (−0.683 − 4.74i)23-s + (−2.99 − 4.00i)25-s + (−0.136 − 0.367i)27-s + (−4.38 + 2.00i)29-s + (−0.475 − 3.30i)31-s + ⋯
L(s)  = 1  + (−1.11 + 0.835i)3-s + (0.447 − 0.894i)5-s + (0.0330 − 0.461i)7-s + (0.266 − 0.907i)9-s + (−0.834 + 1.29i)11-s + (−1.92 + 0.137i)13-s + (0.247 + 1.37i)15-s + (0.463 − 1.24i)17-s + (−0.257 + 0.563i)19-s + (0.349 + 0.543i)21-s + (−0.142 − 0.989i)23-s + (−0.599 − 0.800i)25-s + (−0.0263 − 0.0706i)27-s + (−0.813 + 0.371i)29-s + (−0.0853 − 0.593i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.814 + 0.580i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.814 + 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0468641 - 0.146530i\)
\(L(\frac12)\) \(\approx\) \(0.0468641 - 0.146530i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.00 + 1.99i)T \)
23 \( 1 + (0.683 + 4.74i)T \)
good3 \( 1 + (1.93 - 1.44i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (-0.0873 + 1.22i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (2.76 - 4.30i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (6.94 - 0.497i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (-1.90 + 5.11i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (1.12 - 2.45i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (4.38 - 2.00i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.475 + 3.30i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (3.88 + 7.12i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (9.47 - 2.78i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-1.82 - 2.43i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (8.63 - 8.63i)T - 47iT^{2} \)
53 \( 1 + (-12.1 - 0.867i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (6.30 + 5.46i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (-8.99 + 1.29i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (1.68 + 0.366i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (-1.28 + 0.828i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (7.37 - 2.74i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (0.380 - 0.438i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (5.68 - 3.10i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (0.290 - 2.02i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (4.68 + 2.55i)T + (52.4 + 81.6i)T^{2} \)
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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

−10.40969109396541717587319650851, −9.954643553749460249100798491496, −9.361845784398440400099811659257, −7.82143322603094717956220632133, −6.98606473718091013017459418008, −5.52816410810230204334015931793, −4.93945031535351041037725317440, −4.35966136979764253252684595422, −2.26814558602195788948340882232, −0.10022477898755315199607736475, 1.99277834140949783251731590685, 3.24367932225025803204838335274, 5.34225088248073917623615475183, 5.67437541558191268050845360020, 6.77382548076544375465132995765, 7.45617226038954912422353537684, 8.570067554698641618571581328748, 10.00874576714455283387132796615, 10.56314334230316700546029099080, 11.56642968111365380107291955087

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