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Properties

Label 2-507-13.12-c1-0-1
Degree $2$
Conductor $507$
Sign $-0.246 + 0.969i$
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  + 2.35i·2-s − 3-s − 3.55·4-s + 3.69i·5-s − 2.35i·6-s + 0.801i·7-s − 3.66i·8-s + 9-s − 8.70·10-s + 2.85i·11-s + 3.55·12-s − 1.89·14-s − 3.69i·15-s + 1.52·16-s − 2.93·17-s + 2.35i·18-s + ⋯
L(s)  = 1  + 1.66i·2-s − 0.577·3-s − 1.77·4-s + 1.65i·5-s − 0.962i·6-s + 0.303i·7-s − 1.29i·8-s + 0.333·9-s − 2.75·10-s + 0.859i·11-s + 1.02·12-s − 0.505·14-s − 0.953i·15-s + 0.381·16-s − 0.712·17-s + 0.555i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.246 + 0.969i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.246 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.519852 - 0.668887i\)
\(L(\frac12)\) \(\approx\) \(0.519852 - 0.668887i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 2.35iT - 2T^{2} \)
5 \( 1 - 3.69iT - 5T^{2} \)
7 \( 1 - 0.801iT - 7T^{2} \)
11 \( 1 - 2.85iT - 11T^{2} \)
17 \( 1 + 2.93T + 17T^{2} \)
19 \( 1 + 2.44iT - 19T^{2} \)
23 \( 1 - 7.78T + 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 - 2.34iT - 31T^{2} \)
37 \( 1 + 7.44iT - 37T^{2} \)
41 \( 1 + 0.850iT - 41T^{2} \)
43 \( 1 - 1.61T + 43T^{2} \)
47 \( 1 + 2.44iT - 47T^{2} \)
53 \( 1 + 9.96T + 53T^{2} \)
59 \( 1 - 5.38iT - 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 - 14.3iT - 67T^{2} \)
71 \( 1 - 8.12iT - 71T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 - 5.40T + 79T^{2} \)
83 \( 1 - 7.04iT - 83T^{2} \)
89 \( 1 + 1.13iT - 89T^{2} \)
97 \( 1 + 5.94iT - 97T^{2} \)
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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

−11.29651562022001663872092744697, −10.67730813311162450051522106595, −9.602456298438616768379024647884, −8.678666369497699566881003554996, −7.30939501511148305802414038934, −7.04250116292894370864458278998, −6.30166516976502012968044882466, −5.32653231624079750034862028456, −4.29352358505292305525147263723, −2.65229395028605778009515408795, 0.57744192035166526927365774838, 1.54336874929368138992200971270, 3.23944554141581949554549581463, 4.48571502182304183484283474871, 4.99029319833712095605959702493, 6.30113094406481123309231254904, 7.987522393159066500312020962573, 8.944405568967205516879457782556, 9.452069474190658940300417869336, 10.59007404729874233274310227969

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