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Properties

Label 2-460-115.103-c1-0-4
Degree $2$
Conductor $460$
Sign $0.835 - 0.549i$
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  + (1.50 − 1.12i)3-s + (−0.398 + 2.20i)5-s + (−0.218 + 3.05i)7-s + (0.147 − 0.500i)9-s + (0.865 − 1.34i)11-s + (4.66 − 0.333i)13-s + (1.87 + 3.75i)15-s + (−0.344 + 0.924i)17-s + (−3.35 + 7.34i)19-s + (3.11 + 4.84i)21-s + (0.856 − 4.71i)23-s + (−4.68 − 1.75i)25-s + (1.62 + 4.35i)27-s + (6.54 − 2.98i)29-s + (−0.753 − 5.24i)31-s + ⋯
L(s)  = 1  + (0.867 − 0.649i)3-s + (−0.178 + 0.983i)5-s + (−0.0826 + 1.15i)7-s + (0.0490 − 0.166i)9-s + (0.260 − 0.406i)11-s + (1.29 − 0.0925i)13-s + (0.484 + 0.969i)15-s + (−0.0836 + 0.224i)17-s + (−0.769 + 1.68i)19-s + (0.678 + 1.05i)21-s + (0.178 − 0.983i)23-s + (−0.936 − 0.350i)25-s + (0.312 + 0.838i)27-s + (1.21 − 0.555i)29-s + (−0.135 − 0.941i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.835 - 0.549i$
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: \((2,\ 460,\ (\ :1/2),\ 0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73343 + 0.518406i\)
\(L(\frac12)\) \(\approx\) \(1.73343 + 0.518406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.398 - 2.20i)T \)
23 \( 1 + (-0.856 + 4.71i)T \)
good3 \( 1 + (-1.50 + 1.12i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (0.218 - 3.05i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (-0.865 + 1.34i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-4.66 + 0.333i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (0.344 - 0.924i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (3.35 - 7.34i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-6.54 + 2.98i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.753 + 5.24i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (3.09 + 5.66i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (-8.48 + 2.49i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (1.30 + 1.73i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (2.62 - 2.62i)T - 47iT^{2} \)
53 \( 1 + (-3.29 - 0.235i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (7.42 + 6.43i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (6.05 - 0.871i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (8.47 + 1.84i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (-6.81 + 4.38i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (8.77 - 3.27i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (-8.49 + 9.80i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (12.1 - 6.64i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (1.23 - 8.55i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (2.29 + 1.25i)T + (52.4 + 81.6i)T^{2} \)
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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

−11.07338533995890504830777404997, −10.37735909983505307460401156028, −9.026452137830534085790578634167, −8.374090527568466298324984124492, −7.71244670584045054212417292995, −6.36041878034742911358726006637, −5.92139681467188817448822685326, −3.96158445943664046838806623144, −2.90325483565341156131389559788, −1.95559990180118696547739996909, 1.16344383278713807445080226733, 3.15160277270832184805453298276, 4.15523901548098243078785682060, 4.79184833091666669699414660354, 6.41442093111433277626447981521, 7.43839130413564856942309255815, 8.654513401089509256180550413860, 8.948045571177223658276136283526, 9.952464034875503713975349123199, 10.84147574518105459826113544006

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