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Properties

Label 2-460-115.103-c1-0-0
Degree $2$
Conductor $460$
Sign $-0.775 + 0.631i$
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  + (−2.30 + 1.72i)3-s + (−1.13 + 1.92i)5-s + (−0.304 + 4.26i)7-s + (1.49 − 5.10i)9-s + (−1.16 + 1.81i)11-s + (0.287 − 0.0205i)13-s + (−0.717 − 6.41i)15-s + (0.935 − 2.50i)17-s + (0.691 − 1.51i)19-s + (−6.66 − 10.3i)21-s + (1.30 + 4.61i)23-s + (−2.43 − 4.36i)25-s + (2.34 + 6.27i)27-s + (−6.22 + 2.84i)29-s + (−1.15 − 8.06i)31-s + ⋯
L(s)  = 1  + (−1.33 + 0.998i)3-s + (−0.506 + 0.862i)5-s + (−0.115 + 1.61i)7-s + (0.499 − 1.70i)9-s + (−0.352 + 0.548i)11-s + (0.0796 − 0.00569i)13-s + (−0.185 − 1.65i)15-s + (0.226 − 0.608i)17-s + (0.158 − 0.347i)19-s + (−1.45 − 2.26i)21-s + (0.271 + 0.962i)23-s + (−0.486 − 0.873i)25-s + (0.450 + 1.20i)27-s + (−1.15 + 0.527i)29-s + (−0.208 − 1.44i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.775 + 0.631i$
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: \((2,\ 460,\ (\ :1/2),\ -0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146215 - 0.411090i\)
\(L(\frac12)\) \(\approx\) \(0.146215 - 0.411090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 - 1.92i)T \)
23 \( 1 + (-1.30 - 4.61i)T \)
good3 \( 1 + (2.30 - 1.72i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (0.304 - 4.26i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (1.16 - 1.81i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-0.287 + 0.0205i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (-0.935 + 2.50i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (-0.691 + 1.51i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (6.22 - 2.84i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (1.15 + 8.06i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-1.42 - 2.60i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (-7.94 + 2.33i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-0.259 - 0.345i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (2.99 - 2.99i)T - 47iT^{2} \)
53 \( 1 + (12.7 + 0.909i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (-5.71 - 4.95i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (-4.99 + 0.718i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (-10.8 - 2.36i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (-1.24 + 0.798i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-13.1 + 4.89i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (10.6 - 12.2i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (9.87 - 5.38i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (0.990 - 6.89i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-0.715 - 0.390i)T + (52.4 + 81.6i)T^{2} \)
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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

−11.29422655807771619640345434025, −11.10097652644376456482514209347, −9.723855968304980224919770930371, −9.410160704703553936640502408770, −7.890038528899435162687582744948, −6.75369203208854431896524265833, −5.73663292613557320028619674292, −5.17602524318284209422139884730, −3.92023991939303830237538627931, −2.62281798523426439031129742253, 0.34372392117219228345830334981, 1.36577758034077832141473768339, 3.77009221003203371689163728304, 4.84344667957330711711093889772, 5.83241761386631191073368625557, 6.83234810426527930355486555086, 7.60333932971247301124795300743, 8.349791000559915030415746453462, 9.865649823196983438437089354084, 10.96419619979968566171093695966

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