Properties

Label 2-460-23.3-c1-0-3
Degree $2$
Conductor $460$
Sign $0.882 - 0.470i$
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  + (0.127 − 0.279i)3-s + (0.654 − 0.755i)5-s + (−2.94 + 1.89i)7-s + (1.90 + 2.19i)9-s + (−0.139 + 0.967i)11-s + (5.19 + 3.33i)13-s + (−0.127 − 0.279i)15-s + (1.70 + 0.499i)17-s + (7.80 − 2.29i)19-s + (0.153 + 1.06i)21-s + (−4.53 − 1.55i)23-s + (−0.142 − 0.989i)25-s + (1.74 − 0.511i)27-s + (2.15 + 0.631i)29-s + (−1.59 − 3.48i)31-s + ⋯
L(s)  = 1  + (0.0737 − 0.161i)3-s + (0.292 − 0.337i)5-s + (−1.11 + 0.715i)7-s + (0.634 + 0.731i)9-s + (−0.0419 + 0.291i)11-s + (1.43 + 0.925i)13-s + (−0.0329 − 0.0721i)15-s + (0.412 + 0.121i)17-s + (1.79 − 0.525i)19-s + (0.0334 + 0.232i)21-s + (−0.945 − 0.324i)23-s + (−0.0284 − 0.197i)25-s + (0.335 − 0.0984i)27-s + (0.399 + 0.117i)29-s + (−0.285 − 0.625i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.882 - 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43481 + 0.358414i\)
\(L(\frac12)\) \(\approx\) \(1.43481 + 0.358414i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.53 + 1.55i)T \)
good3 \( 1 + (-0.127 + 0.279i)T + (-1.96 - 2.26i)T^{2} \)
7 \( 1 + (2.94 - 1.89i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (0.139 - 0.967i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-5.19 - 3.33i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-1.70 - 0.499i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-7.80 + 2.29i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-2.15 - 0.631i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (1.59 + 3.48i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (2.03 + 2.34i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (1.16 - 1.34i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (3.43 - 7.51i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 - 5.00T + 47T^{2} \)
53 \( 1 + (8.36 - 5.37i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-0.706 - 0.454i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (0.0599 + 0.131i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-0.387 - 2.69i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (-1.03 - 7.22i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-5.82 + 1.70i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (7.55 + 4.85i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (9.01 + 10.4i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-6.41 + 14.0i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (6.58 - 7.59i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.18661819290427874466446584792, −9.990817300799811773275935980582, −9.433259504648254527272029699210, −8.525656244494482116760390474061, −7.42886374129829752177853941740, −6.39947267535430412034966243744, −5.58505692973954074064231011859, −4.31492493709289727649278196244, −3.01547901175112443587170738297, −1.57887577450139976962937135272, 1.07492571396184527172714474383, 3.37281653977510473120607855681, 3.62236908203044377137318615640, 5.47175875234737932465002062070, 6.34739491026933210293285788965, 7.19129977519442898983240396434, 8.253652417174415669903725645134, 9.495404787536324212155113067346, 10.03427320782212044926995934690, 10.73296029065094527510056369385

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