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Properties

Label 2-513-171.106-c1-0-13
Degree $2$
Conductor $513$
Sign $0.893 + 0.449i$
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  + (0.269 + 0.466i)2-s + (0.854 − 1.48i)4-s + 0.947·5-s + (1.18 − 2.05i)7-s + 1.99·8-s + (0.255 + 0.442i)10-s + (−1.76 + 3.05i)11-s + (0.514 − 0.890i)13-s + 1.27·14-s + (−1.17 − 2.02i)16-s + (0.347 − 0.602i)17-s + (2.46 − 3.59i)19-s + (0.809 − 1.40i)20-s − 1.90·22-s + (1.69 − 2.92i)23-s + ⋯
L(s)  = 1  + (0.190 + 0.330i)2-s + (0.427 − 0.740i)4-s + 0.423·5-s + (0.447 − 0.775i)7-s + 0.706·8-s + (0.0807 + 0.139i)10-s + (−0.532 + 0.922i)11-s + (0.142 − 0.246i)13-s + 0.341·14-s + (−0.292 − 0.506i)16-s + (0.0843 − 0.146i)17-s + (0.565 − 0.824i)19-s + (0.181 − 0.313i)20-s − 0.405·22-s + (0.352 − 0.610i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.893 + 0.449i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.893 + 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90418 - 0.452136i\)
\(L(\frac12)\) \(\approx\) \(1.90418 - 0.452136i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.46 + 3.59i)T \)
good2 \( 1 + (-0.269 - 0.466i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.947T + 5T^{2} \)
7 \( 1 + (-1.18 + 2.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.76 - 3.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.514 + 0.890i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.347 + 0.602i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.69 + 2.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.53T + 29T^{2} \)
31 \( 1 + (-4.48 - 7.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.345T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + (-2.10 - 3.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + (-3.33 - 5.77i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 - 3.47T + 61T^{2} \)
67 \( 1 + (-1.02 + 1.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.75 - 3.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.57 - 7.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.87 + 11.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.41 - 4.18i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.902 - 1.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.02 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.71480862739460641560012037305, −10.08235398615748571187432585132, −9.244180938906684550029474940405, −7.79061927896049811139826919538, −7.19261321241978540254388836473, −6.21845667240938144109384188051, −5.16195412309633677402216292262, −4.44172135590977276566647054318, −2.62838595612889126174248325165, −1.26784536797452869275781464765, 1.84311333485773864019736599040, 2.92098639821146920736236352667, 4.02694990294186297381935825278, 5.44469105846624312678046485887, 6.15920535240326965114823353884, 7.58752999470871170575368955488, 8.152698249156252422700771194335, 9.164615729998391175658287519058, 10.17536571499076364771759938415, 11.33519457037998789956807340939

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