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Properties

Label 2-507-39.32-c1-0-13
Degree $2$
Conductor $507$
Sign $0.916 - 0.399i$
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 + (−1.60 + 0.650i)3-s + (0.232 − 0.133i)4-s + (1.06 + 1.06i)5-s + (2.08 − 1.57i)6-s + (−0.366 + 1.36i)7-s + (1.84 − 1.84i)8-s + (2.15 − 2.08i)9-s + (−1.96 − 1.13i)10-s + (−1.06 − 3.97i)11-s + (−0.285 + 0.366i)12-s − 2.12i·14-s + (−2.40 − 1.01i)15-s + (−2.23 + 3.86i)16-s + (−2.51 − 4.36i)17-s + (−2.31 + 3.87i)18-s + ⋯
L(s)  = 1  + (−1.02 + 0.275i)2-s + (−0.926 + 0.375i)3-s + (0.116 − 0.0669i)4-s + (0.476 + 0.476i)5-s + (0.849 − 0.641i)6-s + (−0.138 + 0.516i)7-s + (0.652 − 0.652i)8-s + (0.717 − 0.696i)9-s + (−0.621 − 0.358i)10-s + (−0.321 − 1.19i)11-s + (−0.0823 + 0.105i)12-s − 0.569i·14-s + (−0.620 − 0.262i)15-s + (−0.558 + 0.966i)16-s + (−0.611 − 1.05i)17-s + (−0.546 + 0.913i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.916 - 0.399i$
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: \((2,\ 507,\ (\ :1/2),\ 0.916 - 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561303 + 0.116953i\)
\(L(\frac12)\) \(\approx\) \(0.561303 + 0.116953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 - 0.650i)T \)
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

−10.79223158374223657960057829009, −9.973059537417069492220687684941, −9.323318220253235222340579629436, −8.462572128343446032688050737120, −7.32929989157006025081557375185, −6.40890587083136616057635038400, −5.60406148843847821954800400757, −4.40809699269477077122043360799, −2.88517853406774644525367362005, −0.75728726497467023247403713233, 0.983634087168849504202564718852, 2.09096545732883971089114848175, 4.43171567307234226472407754072, 5.15686631381056376594741499013, 6.36801986372368586872849914739, 7.37953200151122792446333508474, 8.155841564941618601880883036524, 9.349035895468912097547800320848, 10.01275702852515209963199480804, 10.62083182233852951364151609303

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