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Properties

Label 2-507-39.20-c1-0-34
Degree $2$
Conductor $507$
Sign $0.0532 + 0.998i$
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  + (−0.258 − 0.965i)2-s + (1.72 − 0.158i)3-s + (0.866 − 0.5i)4-s + (1.41 − 1.41i)5-s + (−0.599 − 1.62i)6-s + (1.36 + 0.366i)7-s + (−2.12 − 2.12i)8-s + (2.94 − 0.548i)9-s + (−1.73 − 1.00i)10-s + (−3.86 + 1.03i)11-s + (1.41 − i)12-s − 1.41i·14-s + (2.21 − 2.66i)15-s + (−0.500 + 0.866i)16-s + (−1.29 − 2.70i)18-s + (−0.366 + 1.36i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.995 − 0.0917i)3-s + (0.433 − 0.250i)4-s + (0.632 − 0.632i)5-s + (−0.244 − 0.663i)6-s + (0.516 + 0.138i)7-s + (−0.749 − 0.749i)8-s + (0.983 − 0.182i)9-s + (−0.547 − 0.316i)10-s + (−1.16 + 0.312i)11-s + (0.408 − 0.288i)12-s − 0.377i·14-s + (0.571 − 0.687i)15-s + (−0.125 + 0.216i)16-s + (−0.304 − 0.638i)18-s + (−0.0839 + 0.313i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0532 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0532 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0532 + 0.998i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0532 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60989 - 1.52634i\)
\(L(\frac12)\) \(\approx\) \(1.60989 - 1.52634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.158i)T \)
13 \( 1 \)
good2 \( 1 + (0.258 + 0.965i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-1.41 + 1.41i)T - 5iT^{2} \)
7 \( 1 + (-1.36 - 0.366i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.86 - 1.03i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.366 - 1.36i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.24 - 7.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.44 + 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5 - 5i)T + 31iT^{2} \)
37 \( 1 + (0.366 + 1.36i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.517 - 1.93i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.19 + 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.82 - 2.82i)T + 47iT^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + (-1.03 + 3.86i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.83 - 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.86 - 1.03i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (-5.65 + 5.65i)T - 83iT^{2} \)
89 \( 1 + (-13.5 + 3.62i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.56 - 9.56i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.47430852111647766675393901135, −9.826692906114303899631515904808, −9.146290087465620764100946723981, −8.102522427456711743024612237124, −7.32431470722043487060075339609, −5.96578220092807610754031865352, −4.95313683619028262218183016276, −3.48869608189476946922834410382, −2.28911434558136776584825524646, −1.50976118927658857287571405903, 2.26801293244161470002201143322, 2.85600053977117600025027052571, 4.43283842837127838060340654687, 5.80886917038310738518441044627, 6.69922083817348296200127550733, 7.70936393567065099160356605023, 8.173643020860633354562811727859, 9.113566964111884196941314064795, 10.31329863521568441326926830896, 10.76772938578693892334634498242

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