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Properties

Label 2-513-171.49-c1-0-13
Degree $2$
Conductor $513$
Sign $0.980 - 0.195i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
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  + 2.60·2-s + 4.77·4-s + (1.43 + 2.49i)5-s + (−1.80 − 3.12i)7-s + 7.22·8-s + (3.74 + 6.48i)10-s + (0.154 + 0.267i)11-s − 5.47·13-s + (−4.69 − 8.12i)14-s + 9.26·16-s + (−1.60 + 2.78i)17-s + (2.06 − 3.83i)19-s + (6.87 + 11.9i)20-s + (0.402 + 0.697i)22-s − 1.19·23-s + ⋯
L(s)  = 1  + 1.84·2-s + 2.38·4-s + (0.643 + 1.11i)5-s + (−0.681 − 1.17i)7-s + 2.55·8-s + (1.18 + 2.05i)10-s + (0.0466 + 0.0807i)11-s − 1.51·13-s + (−1.25 − 2.17i)14-s + 2.31·16-s + (−0.389 + 0.674i)17-s + (0.473 − 0.880i)19-s + (1.53 + 2.66i)20-s + (0.0858 + 0.148i)22-s − 0.249·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.06861 + 0.401195i\)
\(L(\frac12)\) \(\approx\) \(4.06861 + 0.401195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (-2.06 + 3.83i)T \)
good2 \( 1 - 2.60T + 2T^{2} \)
5 \( 1 + (-1.43 - 2.49i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.80 + 3.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.154 - 0.267i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.47T + 13T^{2} \)
17 \( 1 + (1.60 - 2.78i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + 1.19T + 23T^{2} \)
29 \( 1 + (1.54 - 2.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.960 - 1.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.85T + 37T^{2} \)
41 \( 1 + (4.18 + 7.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 + (0.487 - 0.843i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.22 - 3.85i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.131 + 0.228i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.12 - 1.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 1.83T + 67T^{2} \)
71 \( 1 + (-5.72 + 9.91i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.24 - 5.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.01T + 79T^{2} \)
83 \( 1 + (1.71 + 2.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.421 - 0.730i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.44T + 97T^{2} \)
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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

−10.95899185677498110470788115129, −10.43206219045778285261379360656, −9.582843618253184278668006092376, −7.47508368959787548342086233833, −6.93563180705381443671208724871, −6.30879803768198571074803551206, −5.18139481130936270529044476347, −4.14912364709741780458448495913, −3.15555126910313594992548513135, −2.27117854035186496690592849309, 2.06486644492189245784498853070, 2.96013032384756869775217760049, 4.40336186622640113354436103763, 5.26638073857042151207944028798, 5.77128980077584917658813576262, 6.73665361404362429669825067749, 7.966946243969100561311528947555, 9.346638275981242707844399519658, 9.884881500435916482875217010416, 11.48470285496030120563506754474

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