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Properties

Label 2-464-29.24-c1-0-10
Degree $2$
Conductor $464$
Sign $-0.912 + 0.408i$
Analytic cond. $3.70505$
Root an. cond. $1.92485$
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  + (−0.260 − 1.14i)3-s + (−1.85 − 2.32i)5-s + (0.115 + 0.507i)7-s + (1.46 − 0.707i)9-s + (−0.585 − 0.281i)11-s + (−0.444 − 0.214i)13-s + (−2.16 + 2.72i)15-s − 7.42·17-s + (1.47 − 6.46i)19-s + (0.549 − 0.264i)21-s + (−4.74 + 5.95i)23-s + (−0.853 + 3.73i)25-s + (−3.37 − 4.23i)27-s + (−4.56 + 2.85i)29-s + (−3.72 − 4.67i)31-s + ⋯
L(s)  = 1  + (−0.150 − 0.658i)3-s + (−0.828 − 1.03i)5-s + (0.0438 + 0.191i)7-s + (0.489 − 0.235i)9-s + (−0.176 − 0.0849i)11-s + (−0.123 − 0.0593i)13-s + (−0.560 + 0.702i)15-s − 1.79·17-s + (0.338 − 1.48i)19-s + (0.119 − 0.0577i)21-s + (−0.990 + 1.24i)23-s + (−0.170 + 0.747i)25-s + (−0.650 − 0.815i)27-s + (−0.848 + 0.529i)29-s + (−0.669 − 0.839i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $-0.912 + 0.408i$
Analytic conductor: \(3.70505\)
Root analytic conductor: \(1.92485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1/2),\ -0.912 + 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162410 - 0.760759i\)
\(L(\frac12)\) \(\approx\) \(0.162410 - 0.760759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (4.56 - 2.85i)T \)
good3 \( 1 + (0.260 + 1.14i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (1.85 + 2.32i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (-0.115 - 0.507i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (0.585 + 0.281i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (0.444 + 0.214i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + 7.42T + 17T^{2} \)
19 \( 1 + (-1.47 + 6.46i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (4.74 - 5.95i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (3.72 + 4.67i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (-2.23 + 1.07i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 - 7.82T + 41T^{2} \)
43 \( 1 + (0.404 - 0.507i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-7.92 - 3.81i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (0.717 + 0.900i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 5.31T + 59T^{2} \)
61 \( 1 + (2.15 + 9.45i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-4.07 + 1.96i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (12.7 + 6.13i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-5.44 + 6.83i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (-3.71 + 1.79i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (-0.952 + 4.17i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.743 + 0.932i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (2.61 - 11.4i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.99776300677444350056689984198, −9.430752185586944993348464289958, −8.922915627958788897714613812710, −7.76254363311651659212716298374, −7.18020455962660104871593326384, −5.97634312060132025465166505691, −4.77418925389824489925646981059, −3.92719457347048808590704445860, −2.09521803030342607967663268076, −0.47611251103397378609378223756, 2.32200214017695639549209058825, 3.85069132983340065706492128039, 4.35028400240682989491347827162, 5.82219541137133643050771649057, 6.98497727269706942742092393336, 7.63516553577144611125311647265, 8.739227348709638614588196022953, 9.934228432399595703869359830073, 10.61582280136298210064350630735, 11.14645252891245087625577285875

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