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Properties

Label 2-460-460.419-c1-0-48
Degree $2$
Conductor $460$
Sign $0.426 + 0.904i$
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  + (−1.35 − 0.398i)2-s + (1.47 − 1.69i)3-s + (1.68 + 1.08i)4-s + (2.21 + 0.318i)5-s + (−2.67 + 1.71i)6-s + (2.31 − 1.05i)7-s + (−1.85 − 2.13i)8-s + (−0.292 − 2.03i)9-s + (−2.87 − 1.31i)10-s + (4.31 − 1.26i)12-s + (−3.56 + 0.512i)14-s + (3.79 − 3.29i)15-s + (1.66 + 3.63i)16-s + (−0.413 + 2.87i)18-s + (3.37 + 2.92i)20-s + (1.61 − 5.49i)21-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.849 − 0.980i)3-s + (0.841 + 0.540i)4-s + (0.989 + 0.142i)5-s + (−1.09 + 0.701i)6-s + (0.875 − 0.399i)7-s + (−0.654 − 0.755i)8-s + (−0.0973 − 0.677i)9-s + (−0.909 − 0.415i)10-s + (1.24 − 0.365i)12-s + (−0.952 + 0.136i)14-s + (0.980 − 0.849i)15-s + (0.415 + 0.909i)16-s + (−0.0973 + 0.677i)18-s + (0.755 + 0.654i)20-s + (0.351 − 1.19i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.426 + 0.904i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.426 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30064 - 0.824678i\)
\(L(\frac12)\) \(\approx\) \(1.30064 - 0.824678i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.398i)T \)
5 \( 1 + (-2.21 - 0.318i)T \)
23 \( 1 + (2.01 - 4.35i)T \)
good3 \( 1 + (-1.47 + 1.69i)T + (-0.426 - 2.96i)T^{2} \)
7 \( 1 + (-2.31 + 1.05i)T + (4.58 - 5.29i)T^{2} \)
11 \( 1 + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (6.72 - 4.32i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (4.41 - 30.6i)T^{2} \)
37 \( 1 + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (-1.28 + 8.96i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (7.21 + 6.24i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (38.6 + 44.5i)T^{2} \)
61 \( 1 + (-11.7 + 10.2i)T + (8.68 - 60.3i)T^{2} \)
67 \( 1 + (2.01 - 6.85i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (-59.7 - 38.3i)T^{2} \)
73 \( 1 + (-30.3 - 66.4i)T^{2} \)
79 \( 1 + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (5.48 - 0.789i)T + (79.6 - 23.3i)T^{2} \)
89 \( 1 + (11.6 + 10.1i)T + (12.6 + 88.0i)T^{2} \)
97 \( 1 + (-93.0 - 27.3i)T^{2} \)
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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

−10.78464058326776340358397746209, −9.880580351226610639401798294831, −8.985782358597846893456187761194, −8.218774791020277911281205132361, −7.40739733134322118478282735370, −6.72965687461107507653355693432, −5.39144550977823773837914413866, −3.45765844463485060426276531118, −2.13026369416377010702261029925, −1.48645443346702402839611542542, 1.77489965703195827107508249706, 2.86782783658162842474425540125, 4.56292275204912063799809641265, 5.59625664319041123060962555367, 6.64324325231418918763749110206, 8.117271336131635347281999277764, 8.527027007054941585542612532288, 9.582848703222671636770684035550, 9.850853658232860267728389712726, 10.87895560549444853058998737198

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