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Properties

Label 2-460-115.107-c1-0-2
Degree $2$
Conductor $460$
Sign $0.304 - 0.952i$
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  + (−1.70 − 0.637i)3-s + (−0.291 + 2.21i)5-s + (3.58 − 0.780i)7-s + (0.248 + 0.215i)9-s + (−4.20 + 0.603i)11-s + (−0.264 + 1.21i)13-s + (1.91 − 3.60i)15-s + (−2.65 + 4.85i)17-s + (7.45 + 2.19i)19-s + (−6.63 − 0.953i)21-s + (1.97 + 4.37i)23-s + (−4.83 − 1.29i)25-s + (2.33 + 4.27i)27-s + (2.02 + 6.87i)29-s + (2.42 − 5.31i)31-s + ⋯
L(s)  = 1  + (−0.986 − 0.368i)3-s + (−0.130 + 0.991i)5-s + (1.35 − 0.294i)7-s + (0.0827 + 0.0717i)9-s + (−1.26 + 0.182i)11-s + (−0.0734 + 0.337i)13-s + (0.493 − 0.930i)15-s + (−0.642 + 1.17i)17-s + (1.71 + 0.502i)19-s + (−1.44 − 0.208i)21-s + (0.411 + 0.911i)23-s + (−0.966 − 0.258i)25-s + (0.449 + 0.823i)27-s + (0.375 + 1.27i)29-s + (0.436 − 0.954i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.304 - 0.952i$
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: \((2,\ 460,\ (\ :1/2),\ 0.304 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.722239 + 0.527314i\)
\(L(\frac12)\) \(\approx\) \(0.722239 + 0.527314i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.291 - 2.21i)T \)
23 \( 1 + (-1.97 - 4.37i)T \)
good3 \( 1 + (1.70 + 0.637i)T + (2.26 + 1.96i)T^{2} \)
7 \( 1 + (-3.58 + 0.780i)T + (6.36 - 2.90i)T^{2} \)
11 \( 1 + (4.20 - 0.603i)T + (10.5 - 3.09i)T^{2} \)
13 \( 1 + (0.264 - 1.21i)T + (-11.8 - 5.40i)T^{2} \)
17 \( 1 + (2.65 - 4.85i)T + (-9.19 - 14.3i)T^{2} \)
19 \( 1 + (-7.45 - 2.19i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-2.02 - 6.87i)T + (-24.3 + 15.6i)T^{2} \)
31 \( 1 + (-2.42 + 5.31i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-0.219 - 3.07i)T + (-36.6 + 5.26i)T^{2} \)
41 \( 1 + (-1.02 - 1.17i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (3.93 - 10.5i)T + (-32.4 - 28.1i)T^{2} \)
47 \( 1 + (-1.35 - 1.35i)T + 47iT^{2} \)
53 \( 1 + (1.04 + 4.82i)T + (-48.2 + 22.0i)T^{2} \)
59 \( 1 + (7.60 + 11.8i)T + (-24.5 + 53.6i)T^{2} \)
61 \( 1 + (3.61 + 1.65i)T + (39.9 + 46.1i)T^{2} \)
67 \( 1 + (-8.22 - 10.9i)T + (-18.8 + 64.2i)T^{2} \)
71 \( 1 + (-2.02 + 14.0i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (5.82 - 3.18i)T + (39.4 - 61.4i)T^{2} \)
79 \( 1 + (-2.48 + 1.59i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (1.99 - 0.142i)T + (82.1 - 11.8i)T^{2} \)
89 \( 1 + (-3.10 - 6.80i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (4.48 + 0.320i)T + (96.0 + 13.8i)T^{2} \)
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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

−11.20987220751488667277769106140, −10.72759680809363636029527363140, −9.715641160040492875965894917523, −8.139701510356395881243805364962, −7.56926383800223345376054908524, −6.59666430727964538501514144465, −5.57837109869053404712758345578, −4.74471441857126558074061748326, −3.19390435220959832233225565972, −1.57926330333803315547637229822, 0.65873907243216237865340066635, 2.57378962627471597395421958827, 4.66129820454741472314857182713, 5.04455584641463651965748235055, 5.68608232126221842169110883077, 7.35173375631676259896230840078, 8.189454443213627922969536455602, 8.997418521168931184744451667090, 10.19257801048527534584705821710, 11.02286948530236775244816503020

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