Properties

Label 2-460-23.18-c1-0-4
Degree $2$
Conductor $460$
Sign $0.993 + 0.112i$
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  + (0.329 − 0.380i)3-s + (−0.142 + 0.989i)5-s + (−0.847 − 1.85i)7-s + (0.390 + 2.71i)9-s + (4.95 − 1.45i)11-s + (0.773 − 1.69i)13-s + (0.329 + 0.380i)15-s + (0.687 − 0.441i)17-s + (5.37 + 3.45i)19-s + (−0.983 − 0.288i)21-s + (4.01 − 2.62i)23-s + (−0.959 − 0.281i)25-s + (2.43 + 1.56i)27-s + (−1.82 + 1.17i)29-s + (−2.33 − 2.69i)31-s + ⋯
L(s)  = 1  + (0.190 − 0.219i)3-s + (−0.0636 + 0.442i)5-s + (−0.320 − 0.701i)7-s + (0.130 + 0.906i)9-s + (1.49 − 0.438i)11-s + (0.214 − 0.469i)13-s + (0.0850 + 0.0981i)15-s + (0.166 − 0.107i)17-s + (1.23 + 0.792i)19-s + (−0.214 − 0.0630i)21-s + (0.837 − 0.546i)23-s + (−0.191 − 0.0563i)25-s + (0.467 + 0.300i)27-s + (−0.339 + 0.218i)29-s + (−0.418 − 0.483i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.993 + 0.112i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58518 - 0.0892890i\)
\(L(\frac12)\) \(\approx\) \(1.58518 - 0.0892890i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (-4.01 + 2.62i)T \)
good3 \( 1 + (-0.329 + 0.380i)T + (-0.426 - 2.96i)T^{2} \)
7 \( 1 + (0.847 + 1.85i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-4.95 + 1.45i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-0.773 + 1.69i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.687 + 0.441i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-5.37 - 3.45i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (1.82 - 1.17i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (2.33 + 2.69i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (0.0791 + 0.550i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (-0.207 + 1.44i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (6.92 - 7.99i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 - 1.60T + 47T^{2} \)
53 \( 1 + (2.53 + 5.55i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (2.28 - 5.00i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (0.676 + 0.780i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-4.15 - 1.21i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (0.396 + 0.116i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (5.14 + 3.30i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (4.50 - 9.85i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (2.02 + 14.0i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (8.47 - 9.78i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (2.60 - 18.0i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.03225337464687917586205305615, −10.19410053476430970026395935693, −9.313363779174760592485412696267, −8.199213565952537524389713441462, −7.33959038472900328620342759883, −6.56283819843668554145461055640, −5.39881965935053457804393843865, −4.00765458242682506291246334354, −3.07194414602431296857114043491, −1.33152744911835822783556703826, 1.36591035411896526019243091749, 3.17458932373806496592572073821, 4.14435028875464908332149718966, 5.35712101125630921913746778720, 6.47956839593417822708784106825, 7.25556433320997246633649072429, 8.803662831197264527159382493046, 9.190700498207501199396558638299, 9.815011626333114357009286723007, 11.33348379505762691267532829246

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