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Properties

Label 2-507-39.32-c1-0-8
Degree $2$
Conductor $507$
Sign $-0.163 - 0.986i$
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  + (1.45 − 0.389i)2-s + (0.239 + 1.71i)3-s + (0.232 − 0.133i)4-s + (−1.06 − 1.06i)5-s + (1.01 + 2.40i)6-s + (−0.366 + 1.36i)7-s + (−1.84 + 1.84i)8-s + (−2.88 + 0.820i)9-s + (−1.96 − 1.13i)10-s + (1.06 + 3.97i)11-s + (0.285 + 0.366i)12-s + 2.12i·14-s + (1.57 − 2.08i)15-s + (−2.23 + 3.86i)16-s + (2.51 + 4.36i)17-s + (−3.87 + 2.31i)18-s + ⋯
L(s)  = 1  + (1.02 − 0.275i)2-s + (0.138 + 0.990i)3-s + (0.116 − 0.0669i)4-s + (−0.476 − 0.476i)5-s + (0.415 + 0.980i)6-s + (−0.138 + 0.516i)7-s + (−0.652 + 0.652i)8-s + (−0.961 + 0.273i)9-s + (−0.621 − 0.358i)10-s + (0.321 + 1.19i)11-s + (0.0823 + 0.105i)12-s + 0.569i·14-s + (0.405 − 0.537i)15-s + (−0.558 + 0.966i)16-s + (0.611 + 1.05i)17-s + (−0.913 + 0.546i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.163 - 0.986i$
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: \((2,\ 507,\ (\ :1/2),\ -0.163 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17635 + 1.38750i\)
\(L(\frac12)\) \(\approx\) \(1.17635 + 1.38750i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.239 - 1.71i)T \)
13 \( 1 \)
good2 \( 1 + (-1.45 + 0.389i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.06 + 1.06i)T + 5iT^{2} \)
7 \( 1 + (0.366 - 1.36i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.06 - 3.97i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.51 - 4.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.73 - i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.20 + 3.58i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.46 + 2.46i)T - 31iT^{2} \)
37 \( 1 + (-5.23 + 1.40i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.42 - 1.45i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.90 + 1.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.25 + 4.25i)T - 47iT^{2} \)
53 \( 1 - 0.779iT - 53T^{2} \)
59 \( 1 + (-2.90 - 0.779i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.53 - 5.73i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.779 + 2.90i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.901 - 0.901i)T + 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (2.90 + 2.90i)T + 83iT^{2} \)
89 \( 1 + (-2.41 - 9.01i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.63 + 0.437i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.50536823571280956214810108644, −10.23966073788772812155003696440, −9.449909728911117323407350725741, −8.612381137502497429683159184975, −7.72383302577349059036198348169, −6.02450263411121252227721323084, −5.25125354013980950394506854291, −4.27807264469431567537230801161, −3.73030492021070528036463167990, −2.40366690777779871382732690868, 0.796253900134437259265293530987, 3.05290954055775619305096693903, 3.61061634248376991698046208201, 5.15454028055749951183012365732, 5.99603475215007394454943284758, 6.97780398592058894615630820970, 7.53534060480870806986633339642, 8.759370404679563521108433437911, 9.681796048600629115000054129884, 11.13459446361007619147815593747

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