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Properties

Label 2-507-39.32-c1-0-16
Degree $2$
Conductor $507$
Sign $0.824 - 0.565i$
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  + (−2.49 + 0.668i)2-s + (0.116 + 1.72i)3-s + (4.03 − 2.33i)4-s + (−0.624 − 0.624i)5-s + (−1.44 − 4.23i)6-s + (0.435 − 1.62i)7-s + (−4.86 + 4.86i)8-s + (−2.97 + 0.401i)9-s + (1.97 + 1.14i)10-s + (0.0927 + 0.346i)11-s + (4.49 + 6.70i)12-s + 4.34i·14-s + (1.00 − 1.15i)15-s + (4.20 − 7.29i)16-s + (−1.31 − 2.27i)17-s + (7.14 − 2.98i)18-s + ⋯
L(s)  = 1  + (−1.76 + 0.472i)2-s + (0.0669 + 0.997i)3-s + (2.01 − 1.16i)4-s + (−0.279 − 0.279i)5-s + (−0.589 − 1.72i)6-s + (0.164 − 0.613i)7-s + (−1.71 + 1.71i)8-s + (−0.991 + 0.133i)9-s + (0.624 + 0.360i)10-s + (0.0279 + 0.104i)11-s + (1.29 + 1.93i)12-s + 1.15i·14-s + (0.260 − 0.297i)15-s + (1.05 − 1.82i)16-s + (−0.318 − 0.551i)17-s + (1.68 − 0.703i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.824 - 0.565i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.824 - 0.565i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.564563 + 0.175012i\)
\(L(\frac12)\) \(\approx\) \(0.564563 + 0.175012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.116 - 1.72i)T \)
13 \( 1 \)
good2 \( 1 + (2.49 - 0.668i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.624 + 0.624i)T + 5iT^{2} \)
7 \( 1 + (-0.435 + 1.62i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.0927 - 0.346i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.31 + 2.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.90 - 1.58i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.70 + 4.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.24 - 4.18i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.27 - 1.27i)T - 31iT^{2} \)
37 \( 1 + (3.33 - 0.894i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-6.68 + 1.79i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.825 + 0.476i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.33 + 5.33i)T - 47iT^{2} \)
53 \( 1 - 9.69iT - 53T^{2} \)
59 \( 1 + (-9.95 - 2.66i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.43 + 4.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.25 - 12.1i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.832 + 3.10i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.246 + 0.246i)T + 73iT^{2} \)
79 \( 1 + 3.68T + 79T^{2} \)
83 \( 1 + (8.47 + 8.47i)T + 83iT^{2} \)
89 \( 1 + (1.77 + 6.61i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-5.11 - 1.36i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.49410892463238390517425187565, −10.11811799185882140536024834580, −9.069350094693824918453837623039, −8.596286060086469145024051624786, −7.61797291370386769848551699673, −6.80245205564414179608609747234, −5.53593026862762296989031043641, −4.36915374621764754447063819172, −2.75964444919661153376619453306, −0.816674428292540503400727423591, 1.03478059231956552484473105839, 2.28699963950553822701778491412, 3.26789947474968707083353485518, 5.59629345998732211398418226552, 6.77948873041997832774520122550, 7.48151299971136171506542627379, 8.206082626657161078501726958690, 8.997546607967832339590748736623, 9.719739449419776342547907070334, 10.95398311958809176930347381153

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