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Properties

Label 2-460-23.2-c1-0-5
Degree $2$
Conductor $460$
Sign $0.0127 + 0.999i$
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  + (−0.282 − 1.96i)3-s + (0.959 + 0.281i)5-s + (2.71 − 3.13i)7-s + (−0.895 + 0.262i)9-s + (−1.33 + 0.858i)11-s + (2.22 + 2.56i)13-s + (0.282 − 1.96i)15-s + (−0.176 + 0.386i)17-s + (−3.04 − 6.65i)19-s + (−6.91 − 4.44i)21-s + (1.22 + 4.63i)23-s + (0.841 + 0.540i)25-s + (−1.70 − 3.72i)27-s + (0.197 − 0.433i)29-s + (0.289 − 2.01i)31-s + ⋯
L(s)  = 1  + (−0.162 − 1.13i)3-s + (0.429 + 0.125i)5-s + (1.02 − 1.18i)7-s + (−0.298 + 0.0876i)9-s + (−0.402 + 0.258i)11-s + (0.616 + 0.711i)13-s + (0.0728 − 0.506i)15-s + (−0.0427 + 0.0936i)17-s + (−0.697 − 1.52i)19-s + (−1.50 − 0.969i)21-s + (0.254 + 0.966i)23-s + (0.168 + 0.108i)25-s + (−0.327 − 0.717i)27-s + (0.0367 − 0.0804i)29-s + (0.0519 − 0.361i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0127 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0127 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.0127 + 0.999i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.0127 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11175 - 1.09771i\)
\(L(\frac12)\) \(\approx\) \(1.11175 - 1.09771i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (-1.22 - 4.63i)T \)
good3 \( 1 + (0.282 + 1.96i)T + (-2.87 + 0.845i)T^{2} \)
7 \( 1 + (-2.71 + 3.13i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (1.33 - 0.858i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-2.22 - 2.56i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (0.176 - 0.386i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (3.04 + 6.65i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (-0.197 + 0.433i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-0.289 + 2.01i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (0.349 - 0.102i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (6.46 + 1.89i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.0316 - 0.219i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + 0.257T + 47T^{2} \)
53 \( 1 + (2.32 - 2.68i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-4.14 - 4.78i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.776 - 5.39i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (4.25 + 2.73i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (-13.3 - 8.60i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-5.53 - 12.1i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-5.29 - 6.10i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-8.80 + 2.58i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (1.38 + 9.60i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-13.4 - 3.93i)T + (81.6 + 52.4i)T^{2} \)
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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

−11.04775581731017000728181396446, −10.08809173751125422534518226851, −8.887984253450202667777520464561, −7.85350038781552306538724169822, −7.11167606076948734758608283147, −6.48173913171215864024678663599, −5.11221665938260630332743526184, −4.04555653922573421403340181256, −2.18950426768558876560507999759, −1.11804437728564246153861137133, 1.93528467117715374368589808158, 3.42395703383078852062928204851, 4.75615668853062761477121090863, 5.38684932561486243047591481683, 6.27685089842095395957547658240, 8.069763926960157209985768219739, 8.562734653064482163271157580671, 9.563070292342641081165448835674, 10.50921451476487838501567063697, 10.95816700674615893757373106252

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