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Properties

Label 2-507-39.2-c1-0-23
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: $\chi_{507} (80, \cdot )$
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

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

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