Properties

Label 2-460-23.8-c1-0-6
Degree $2$
Conductor $460$
Sign $-0.962 - 0.270i$
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  + (−1.08 − 2.37i)3-s + (−0.654 − 0.755i)5-s + (0.0544 + 0.0349i)7-s + (−2.51 + 2.90i)9-s + (−0.824 − 5.73i)11-s + (−3.15 + 2.02i)13-s + (−1.08 + 2.37i)15-s + (−1.70 + 0.501i)17-s + (3.27 + 0.960i)19-s + (0.0241 − 0.167i)21-s + (−4.09 + 2.49i)23-s + (−0.142 + 0.989i)25-s + (2.11 + 0.621i)27-s + (−1.26 + 0.371i)29-s + (−1.01 + 2.23i)31-s + ⋯
L(s)  = 1  + (−0.627 − 1.37i)3-s + (−0.292 − 0.337i)5-s + (0.0205 + 0.0132i)7-s + (−0.838 + 0.968i)9-s + (−0.248 − 1.72i)11-s + (−0.874 + 0.561i)13-s + (−0.280 + 0.614i)15-s + (−0.414 + 0.121i)17-s + (0.750 + 0.220i)19-s + (0.00525 − 0.0365i)21-s + (−0.854 + 0.519i)23-s + (−0.0284 + 0.197i)25-s + (0.407 + 0.119i)27-s + (−0.234 + 0.0689i)29-s + (−0.183 + 0.400i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.962 - 0.270i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.962 - 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0760430 + 0.552472i\)
\(L(\frac12)\) \(\approx\) \(0.0760430 + 0.552472i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (4.09 - 2.49i)T \)
good3 \( 1 + (1.08 + 2.37i)T + (-1.96 + 2.26i)T^{2} \)
7 \( 1 + (-0.0544 - 0.0349i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (0.824 + 5.73i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (3.15 - 2.02i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (1.70 - 0.501i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (-3.27 - 0.960i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (1.26 - 0.371i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (1.01 - 2.23i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (1.23 - 1.42i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (0.0704 + 0.0812i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (0.298 + 0.653i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 - 0.878T + 47T^{2} \)
53 \( 1 + (7.32 + 4.71i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-12.8 + 8.27i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-0.555 + 1.21i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (-1.53 + 10.6i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (-1.57 + 10.9i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (4.82 + 1.41i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (6.43 - 4.13i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-8.70 + 10.0i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-0.981 - 2.14i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-4.21 - 4.86i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.95416067809024371903982299989, −9.627981993639191077262953162557, −8.462364983111926255053841869164, −7.77544090457859237534979139970, −6.84000064960477398154747127395, −5.95769632650133527614740746588, −5.08958169956476142291433539455, −3.44458904425183650933808770023, −1.84845189212806232157372062991, −0.35925108431913601923553835849, 2.53973384707563534937400676898, 4.03582700587593977636169614992, 4.74357440703907152495576327854, 5.60984074272920761715822870283, 6.97867479147699527073010133873, 7.80268902923116609484328224653, 9.256520522926861721296406313705, 9.970263254324160281776587594183, 10.42302128844435995897942052356, 11.43483604946164158030272162026

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