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Properties

Label 2-507-169.25-c1-0-5
Degree $2$
Conductor $507$
Sign $0.842 - 0.538i$
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  + (−0.239 − 0.0908i)2-s + (0.568 + 0.822i)3-s + (−1.44 − 1.28i)4-s + (−3.85 − 0.468i)5-s + (−0.0612 − 0.248i)6-s + (0.285 + 0.543i)7-s + (0.468 + 0.892i)8-s + (−0.354 + 0.935i)9-s + (0.881 + 0.462i)10-s + (5.04 − 1.91i)11-s + (0.233 − 1.92i)12-s + (0.554 + 3.56i)13-s + (−0.0189 − 0.155i)14-s + (−1.80 − 3.44i)15-s + (0.435 + 3.58i)16-s + (3.94 − 2.06i)17-s + ⋯
L(s)  = 1  + (−0.169 − 0.0642i)2-s + (0.327 + 0.475i)3-s + (−0.723 − 0.641i)4-s + (−1.72 − 0.209i)5-s + (−0.0250 − 0.101i)6-s + (0.107 + 0.205i)7-s + (0.165 + 0.315i)8-s + (−0.118 + 0.311i)9-s + (0.278 + 0.146i)10-s + (1.52 − 0.576i)11-s + (0.0673 − 0.554i)12-s + (0.153 + 0.988i)13-s + (−0.00506 − 0.0416i)14-s + (−0.466 − 0.888i)15-s + (0.108 + 0.896i)16-s + (0.956 − 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.842 - 0.538i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908341 + 0.265706i\)
\(L(\frac12)\) \(\approx\) \(0.908341 + 0.265706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.568 - 0.822i)T \)
13 \( 1 + (-0.554 - 3.56i)T \)
good2 \( 1 + (0.239 + 0.0908i)T + (1.49 + 1.32i)T^{2} \)
5 \( 1 + (3.85 + 0.468i)T + (4.85 + 1.19i)T^{2} \)
7 \( 1 + (-0.285 - 0.543i)T + (-3.97 + 5.76i)T^{2} \)
11 \( 1 + (-5.04 + 1.91i)T + (8.23 - 7.29i)T^{2} \)
17 \( 1 + (-3.94 + 2.06i)T + (9.65 - 13.9i)T^{2} \)
19 \( 1 - 4.79iT - 19T^{2} \)
23 \( 1 - 2.06T + 23T^{2} \)
29 \( 1 + (-1.54 + 4.08i)T + (-21.7 - 19.2i)T^{2} \)
31 \( 1 + (-2.00 - 8.12i)T + (-27.4 + 14.4i)T^{2} \)
37 \( 1 + (0.662 + 2.68i)T + (-32.7 + 17.1i)T^{2} \)
41 \( 1 + (-0.587 + 0.405i)T + (14.5 - 38.3i)T^{2} \)
43 \( 1 + (7.24 + 1.78i)T + (38.0 + 19.9i)T^{2} \)
47 \( 1 + (-5.89 - 6.65i)T + (-5.66 + 46.6i)T^{2} \)
53 \( 1 + (9.45 - 4.96i)T + (30.1 - 43.6i)T^{2} \)
59 \( 1 + (2.75 + 0.334i)T + (57.2 + 14.1i)T^{2} \)
61 \( 1 + (-6.41 - 3.36i)T + (34.6 + 50.2i)T^{2} \)
67 \( 1 + (-4.89 - 5.52i)T + (-8.07 + 66.5i)T^{2} \)
71 \( 1 + (5.61 - 3.87i)T + (25.1 - 66.3i)T^{2} \)
73 \( 1 + (-5.74 + 2.18i)T + (54.6 - 48.4i)T^{2} \)
79 \( 1 + (-6.34 + 5.62i)T + (9.52 - 78.4i)T^{2} \)
83 \( 1 + (-8.63 - 5.96i)T + (29.4 + 77.6i)T^{2} \)
89 \( 1 + 5.00iT - 89T^{2} \)
97 \( 1 + (8.09 - 0.982i)T + (94.1 - 23.2i)T^{2} \)
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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

−11.05947811648046380308487875776, −9.994750944935819030831277979956, −9.000483650238999434554423939901, −8.597019146433670272479313983395, −7.64185929453486549125335551635, −6.39166042338230975551918528446, −5.05250948536949471911183929480, −4.10927681268667028248362914819, −3.52674021343337726714017624403, −1.15641157453343740623285700073, 0.78326847628189398384462174311, 3.20827762699525219674593913123, 3.85396172304740900623126695855, 4.80981373582855151998108890801, 6.65477752964080786341623168374, 7.45433522286397909030609791882, 8.060124726621829142550970238006, 8.769075626447565783700391032180, 9.741815502242494795175899322069, 11.04147249904372818632527480550

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