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Properties

Label 2-507-13.3-c1-0-11
Degree $2$
Conductor $507$
Sign $0.0128 - 0.999i$
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  + (0.866 + 1.5i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 + 1.5i)6-s + (1.73 − 3i)7-s + 1.73·8-s + (−0.499 + 0.866i)9-s + (1.73 + 3i)11-s − 12-s + 6·14-s + (2.49 + 4.33i)16-s + (−3 + 5.19i)17-s − 1.73·18-s + (1.73 − 3i)19-s + 3.46·21-s + (−3 + 5.19i)22-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (0.288 + 0.499i)3-s + (−0.250 + 0.433i)4-s + (−0.353 + 0.612i)6-s + (0.654 − 1.13i)7-s + 0.612·8-s + (−0.166 + 0.288i)9-s + (0.522 + 0.904i)11-s − 0.288·12-s + 1.60·14-s + (0.624 + 1.08i)16-s + (−0.727 + 1.26i)17-s − 0.408·18-s + (0.397 − 0.688i)19-s + 0.755·21-s + (−0.639 + 1.10i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0128 - 0.999i$
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: \((2,\ 507,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72636 + 1.70436i\)
\(L(\frac12)\) \(\approx\) \(1.72636 + 1.70436i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + (-1.73 + 3i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.73 - 3i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 + 3i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 + (3.46 + 6i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.46 + 6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (5.19 - 9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.19 + 9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.73 + 3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 + (-3.46 - 6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.92 - 12i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.90062390626560329053170751235, −10.34970142411798494226516679032, −9.284378420836485645111047704114, −8.105864610898901813475340139241, −7.40846100464523643984076280573, −6.60014161382903242036073866970, −5.47761857973137237889707746894, −4.33232017353456202698221065040, −4.02311220813120766645505299471, −1.82202202495074669595275408454, 1.51288112759546948772989045946, 2.58674421072034485304408973566, 3.52649651814239581667528243385, 4.85090603086031049925575186469, 5.78741246628291012785137877972, 7.06214387103409971680685012792, 8.159046673784514648429178492491, 8.902722682514192041147811383762, 9.931929061707536169344873352416, 11.16326265903685375828083795417

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