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Properties

Label 2-507-13.3-c1-0-4
Degree $2$
Conductor $507$
Sign $-0.668 + 0.743i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (1.17 + 2.04i)2-s + (0.5 + 0.866i)3-s + (−1.77 + 3.07i)4-s − 3.69·5-s + (−1.17 + 2.04i)6-s + (−0.400 + 0.694i)7-s − 3.66·8-s + (−0.499 + 0.866i)9-s + (−4.35 − 7.53i)10-s + (−1.42 − 2.46i)11-s − 3.55·12-s − 1.89·14-s + (−1.84 − 3.19i)15-s + (−0.763 − 1.32i)16-s + (−1.46 + 2.54i)17-s − 2.35·18-s + ⋯
L(s)  = 1  + (0.833 + 1.44i)2-s + (0.288 + 0.499i)3-s + (−0.888 + 1.53i)4-s − 1.65·5-s + (−0.481 + 0.833i)6-s + (−0.151 + 0.262i)7-s − 1.29·8-s + (−0.166 + 0.288i)9-s + (−1.37 − 2.38i)10-s + (−0.429 − 0.744i)11-s − 1.02·12-s − 0.505·14-s + (−0.476 − 0.825i)15-s + (−0.190 − 0.330i)16-s + (−0.356 + 0.617i)17-s − 0.555·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.668 + 0.743i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503792 - 1.13071i\)
\(L(\frac12)\) \(\approx\) \(0.503792 - 1.13071i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-1.17 - 2.04i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 3.69T + 5T^{2} \)
7 \( 1 + (0.400 - 0.694i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.42 + 2.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.46 - 2.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.22 - 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.89 - 6.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.92 + 3.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.34T + 31T^{2} \)
37 \( 1 + (-3.72 - 6.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.425 + 0.736i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.807 + 1.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.44T + 47T^{2} \)
53 \( 1 + 9.96T + 53T^{2} \)
59 \( 1 + (2.69 - 4.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.62 + 11.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.19 - 12.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.06 + 7.03i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 5.40T + 79T^{2} \)
83 \( 1 + 7.04T + 83T^{2} \)
89 \( 1 + (-0.565 - 0.980i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.97 - 5.14i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.51633713862091048501235344911, −10.81894683437230336068562494375, −9.311984255914000322317106038959, −8.191020834943332763886369205665, −7.983601551873093520705236610763, −6.94576455431698888455867938057, −5.85849630009398230571091395445, −4.88694565411790682803627033180, −3.93102796248669076083918032391, −3.28599081964485661315101016379, 0.54441248363324078977583531786, 2.34177057312061585192384103210, 3.33581714242086416098020702785, 4.28106571445404714140991828176, 5.00866379070956067825693471656, 6.82610553447482418659035332052, 7.57424861666838984228204096376, 8.646179140128976044275847462033, 9.706653370816498158113458103884, 11.01276402585211145902374166582

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