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Properties

Label 2-507-13.10-c1-0-17
Degree $2$
Conductor $507$
Sign $0.543 + 0.839i$
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  + (0.358 − 0.207i)2-s + (−0.5 − 0.866i)3-s + (−0.914 + 1.58i)4-s − 2.82i·5-s + (−0.358 − 0.207i)6-s + (2.44 + 1.41i)7-s + 1.58i·8-s + (−0.499 + 0.866i)9-s + (−0.585 − 1.01i)10-s + (1.73 − i)11-s + 1.82·12-s + 1.17·14-s + (−2.44 + 1.41i)15-s + (−1.49 − 2.59i)16-s + (3.82 − 6.63i)17-s + 0.414i·18-s + ⋯
L(s)  = 1  + (0.253 − 0.146i)2-s + (−0.288 − 0.499i)3-s + (−0.457 + 0.791i)4-s − 1.26i·5-s + (−0.146 − 0.0845i)6-s + (0.925 + 0.534i)7-s + 0.560i·8-s + (−0.166 + 0.288i)9-s + (−0.185 − 0.320i)10-s + (0.522 − 0.301i)11-s + 0.527·12-s + 0.313·14-s + (−0.632 + 0.365i)15-s + (−0.374 − 0.649i)16-s + (0.928 − 1.60i)17-s + 0.0976i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.543 + 0.839i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30015 - 0.706585i\)
\(L(\frac12)\) \(\approx\) \(1.30015 - 0.706585i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.358 + 0.207i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 + (-2.44 - 1.41i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.73 + i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.82 + 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.44 - 1.41i)T + (9.5 + 16.4i)T^{2} \)
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 \( 1 + 1.17iT - 31T^{2} \)
37 \( 1 + (-6.63 + 3.82i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.47 + 2.58i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.828 - 1.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-6.63 - 3.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.65 - 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.91 + 3.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.73 + i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.343iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 3.65iT - 83T^{2} \)
89 \( 1 + (12.8 - 7.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.16 + 1.82i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.27480863248940431460413838749, −9.572261554244629050260338410018, −8.891651586605040332997608976073, −8.067993217955035190981589834012, −7.45892165854194879528236783518, −5.79475053397102206646514853134, −5.03616784814114234677622544702, −4.22399579144983153447488275175, −2.64769483546026417808482288239, −1.01731228700102055522114693777, 1.51576210009990524269767261166, 3.48142030319083159872691780325, 4.35293146897602593501727012265, 5.45416061935271138576060437973, 6.30879362759092824517490719819, 7.25167517174378650236544178145, 8.333227402208639292238596577440, 9.667937305533403702470279726829, 10.19865060108369152183238974765, 10.95061351255829358009643163761

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