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Properties

Label 2-513-19.9-c1-0-15
Degree $2$
Conductor $513$
Sign $0.854 - 0.519i$
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  + (0.152 + 0.866i)2-s + (1.15 − 0.419i)4-s + (2.37 + 0.866i)5-s + (1.32 − 2.29i)7-s + (1.41 + 2.45i)8-s + (−0.386 + 2.19i)10-s + (−0.205 − 0.356i)11-s + (−1.12 − 0.943i)13-s + (2.19 + 0.797i)14-s + (−0.0320 + 0.0269i)16-s + (−0.414 − 2.35i)17-s + (−4.29 − 0.725i)19-s + 3.10·20-s + (0.277 − 0.232i)22-s + (0.826 − 0.300i)23-s + ⋯
L(s)  = 1  + (0.107 + 0.612i)2-s + (0.576 − 0.209i)4-s + (1.06 + 0.387i)5-s + (0.501 − 0.868i)7-s + (0.501 + 0.868i)8-s + (−0.122 + 0.693i)10-s + (−0.0620 − 0.107i)11-s + (−0.311 − 0.261i)13-s + (0.585 + 0.213i)14-s + (−0.00802 + 0.00673i)16-s + (−0.100 − 0.570i)17-s + (−0.986 − 0.166i)19-s + 0.694·20-s + (0.0590 − 0.0495i)22-s + (0.172 − 0.0627i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.854 - 0.519i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.854 - 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09519 + 0.587500i\)
\(L(\frac12)\) \(\approx\) \(2.09519 + 0.587500i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.29 + 0.725i)T \)
good2 \( 1 + (-0.152 - 0.866i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-2.37 - 0.866i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.32 + 2.29i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.205 + 0.356i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.12 + 0.943i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.414 + 2.35i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.826 + 0.300i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.06 - 6.01i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.01 + 1.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.69T + 37T^{2} \)
41 \( 1 + (6.53 - 5.48i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-10.6 - 3.86i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.47 - 8.34i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-1.41 + 0.516i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.645 + 3.66i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.35 + 0.858i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.847 - 4.80i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (9.46 + 3.44i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (7.85 - 6.58i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-12.4 + 10.4i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.81 + 4.88i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.22 + 1.03i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.0175 + 0.0994i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.67490125583950519231927058469, −10.39464934898602036575264609781, −9.211383069115441830423131954200, −8.027827458219887027250825007193, −7.15064341792253172653480029688, −6.46535076253209972803483814647, −5.52761470216390343653971272433, −4.57708650372588191247932159619, −2.82455724636598561898407783900, −1.63493508738948190195870748938, 1.79351819145545208793359704609, 2.35570202249088322673591176057, 3.91313051332869522459618681298, 5.22049032022758949956101588599, 6.08622416294914617606271793743, 7.08387089755753109301425802983, 8.333960633310838268859596411774, 9.114887990683385630779099847829, 10.14223846764569375973195245256, 10.74534538933578835374166401798

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