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Properties

Label 2-507-39.5-c1-0-40
Degree $2$
Conductor $507$
Sign $-0.771 - 0.636i$
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  + (−0.540 − 0.540i)2-s + (0.0858 − 1.72i)3-s − 1.41i·4-s + (0.996 + 0.996i)5-s + (−0.981 + 0.888i)6-s + (−1.80 − 1.80i)7-s + (−1.84 + 1.84i)8-s + (−2.98 − 0.296i)9-s − 1.07i·10-s + (−3.35 + 3.35i)11-s + (−2.44 − 0.121i)12-s + 1.94i·14-s + (1.80 − 1.63i)15-s − 0.837·16-s − 5.80·17-s + (1.45 + 1.77i)18-s + ⋯
L(s)  = 1  + (−0.382 − 0.382i)2-s + (0.0495 − 0.998i)3-s − 0.708i·4-s + (0.445 + 0.445i)5-s + (−0.400 + 0.362i)6-s + (−0.681 − 0.681i)7-s + (−0.652 + 0.652i)8-s + (−0.995 − 0.0989i)9-s − 0.340i·10-s + (−1.01 + 1.01i)11-s + (−0.707 − 0.0350i)12-s + 0.520i·14-s + (0.467 − 0.422i)15-s − 0.209·16-s − 1.40·17-s + (0.342 + 0.417i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.771 - 0.636i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165986 + 0.461815i\)
\(L(\frac12)\) \(\approx\) \(0.165986 + 0.461815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0858 + 1.72i)T \)
13 \( 1 \)
good2 \( 1 + (0.540 + 0.540i)T + 2iT^{2} \)
5 \( 1 + (-0.996 - 0.996i)T + 5iT^{2} \)
7 \( 1 + (1.80 + 1.80i)T + 7iT^{2} \)
11 \( 1 + (3.35 - 3.35i)T - 11iT^{2} \)
17 \( 1 + 5.80T + 17T^{2} \)
19 \( 1 + (-2.39 + 2.39i)T - 19iT^{2} \)
23 \( 1 - 3.39T + 23T^{2} \)
29 \( 1 + 6.57iT - 29T^{2} \)
31 \( 1 + (-0.386 + 0.386i)T - 31iT^{2} \)
37 \( 1 + (5.93 + 5.93i)T + 37iT^{2} \)
41 \( 1 + (0.734 + 0.734i)T + 41iT^{2} \)
43 \( 1 - 7.56iT - 43T^{2} \)
47 \( 1 + (-0.243 + 0.243i)T - 47iT^{2} \)
53 \( 1 + 2.07iT - 53T^{2} \)
59 \( 1 + (3.56 - 3.56i)T - 59iT^{2} \)
61 \( 1 - 7.04T + 61T^{2} \)
67 \( 1 + (4.54 - 4.54i)T - 67iT^{2} \)
71 \( 1 + (6.79 + 6.79i)T + 71iT^{2} \)
73 \( 1 + (6.04 + 6.04i)T + 73iT^{2} \)
79 \( 1 - 8.77T + 79T^{2} \)
83 \( 1 + (8.31 + 8.31i)T + 83iT^{2} \)
89 \( 1 + (-9.62 + 9.62i)T - 89iT^{2} \)
97 \( 1 + (1.34 - 1.34i)T - 97iT^{2} \)
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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

−10.38728113710160771827157830801, −9.633854964534891780794601415612, −8.730711590437490341167478019563, −7.44962069425104202142047611890, −6.76353766429349568519596496886, −5.95040946932368843697011168476, −4.74309536673822725317099114750, −2.80537994486316409446654022527, −2.00437191366937395691203114452, −0.30164150988452616019462070827, 2.75237694607163648680288943024, 3.53286573605373569119402192896, 5.00352332391810949045483478786, 5.79828134939649191562905455335, 6.89569146987554728763792258309, 8.252593429284326970881783114104, 8.861967121882180784915834205789, 9.358352845990354144613329835879, 10.42954171842563566890690370184, 11.29390674669068407003711423071

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