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Properties

Label 2-507-13.9-c1-0-15
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  + (1.20 − 2.09i)2-s + (−0.5 + 0.866i)3-s + (−1.91 − 3.31i)4-s + 2.82·5-s + (1.20 + 2.09i)6-s + (1.41 + 2.44i)7-s − 4.41·8-s + (−0.499 − 0.866i)9-s + (3.41 − 5.91i)10-s + (1 − 1.73i)11-s + 3.82·12-s + 6.82·14-s + (−1.41 + 2.44i)15-s + (−1.49 + 2.59i)16-s + (1.82 + 3.16i)17-s − 2.41·18-s + ⋯
L(s)  = 1  + (0.853 − 1.47i)2-s + (−0.288 + 0.499i)3-s + (−0.957 − 1.65i)4-s + 1.26·5-s + (0.492 + 0.853i)6-s + (0.534 + 0.925i)7-s − 1.56·8-s + (−0.166 − 0.288i)9-s + (1.07 − 1.87i)10-s + (0.301 − 0.522i)11-s + 1.10·12-s + 1.82·14-s + (−0.365 + 0.632i)15-s + (−0.374 + 0.649i)16-s + (0.443 + 0.768i)17-s − 0.569·18-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: $\chi_{507} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75013 - 1.72783i\)
\(L(\frac12)\) \(\approx\) \(1.75013 - 1.72783i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 2.09i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 + (-1.41 - 2.44i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.82 - 3.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.82T + 31T^{2} \)
37 \( 1 + (1.82 - 3.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.41 - 9.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.82 + 8.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.343T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-1.82 - 3.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.65 - 8.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.585 - 1.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1 + 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 7.65T + 83T^{2} \)
89 \( 1 + (4.58 - 7.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.82 - 6.63i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.78898545592017312135924629447, −10.13632804048040222591294254077, −9.284226918088972820132613154353, −8.538304927068246590058027959493, −6.45426770035027835327806725168, −5.50473822978601613322554712979, −5.02599982122258267767009437572, −3.71455226739373052324639476623, −2.55457321727063998278556942297, −1.55583427990452834321666908859, 1.74652173092766292815956757184, 3.71828732660315513116485539770, 4.96605272585828715322306163828, 5.56983670656069765919798782972, 6.53573908566459362288316204647, 7.23415362389483094214672477576, 7.929791360054788011283943503418, 9.174369921210459430506877940979, 10.13908414493140565767035017850, 11.23948814289454557864233534378

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