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Properties

Label 2-507-13.4-c1-0-10
Degree $2$
Conductor $507$
Sign $-0.702 - 0.711i$
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  + (2.21 + 1.28i)2-s + (−0.5 + 0.866i)3-s + (2.28 + 3.95i)4-s − 0.561i·5-s + (−2.21 + 1.28i)6-s + (−3.08 + 1.78i)7-s + 6.56i·8-s + (−0.499 − 0.866i)9-s + (0.719 − 1.24i)10-s + (1.73 + i)11-s − 4.56·12-s − 9.12·14-s + (0.486 + 0.280i)15-s + (−3.84 + 6.65i)16-s + (1.28 + 2.21i)17-s − 2.56i·18-s + ⋯
L(s)  = 1  + (1.56 + 0.905i)2-s + (−0.288 + 0.499i)3-s + (1.14 + 1.97i)4-s − 0.251i·5-s + (−0.905 + 0.522i)6-s + (−1.16 + 0.673i)7-s + 2.31i·8-s + (−0.166 − 0.288i)9-s + (0.227 − 0.393i)10-s + (0.522 + 0.301i)11-s − 1.31·12-s − 2.43·14-s + (0.125 + 0.0724i)15-s + (−0.960 + 1.66i)16-s + (0.310 + 0.538i)17-s − 0.603i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.702 - 0.711i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.702 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07476 + 2.57135i\)
\(L(\frac12)\) \(\approx\) \(1.07476 + 2.57135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-2.21 - 1.28i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 0.561iT - 5T^{2} \)
7 \( 1 + (3.08 - 1.78i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.73 - i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.28 - 2.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.972 + 0.561i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.84 + 4.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.56iT - 31T^{2} \)
37 \( 1 + (2.97 + 1.71i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.21 - 1.28i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.219 - 0.379i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (9.63 - 5.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.06 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.379 - 0.219i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (12.1 - 7i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.87iT - 73T^{2} \)
79 \( 1 - 9.56T + 79T^{2} \)
83 \( 1 - 9.12iT - 83T^{2} \)
89 \( 1 + (11.3 + 6.56i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.84 + 2.21i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.74741082256080686065981145479, −10.45523007308899660658466015941, −9.362158317188551052881970822586, −8.440465560962820135094138050145, −7.11363096973566661321757008582, −6.35241716926836993398254501602, −5.66404672099668494598971784913, −4.70579396802107896496553137149, −3.75206209983789581455313749093, −2.78484644899327010629711979436, 1.15060972794886268533283965689, 2.83574566805473002639543772657, 3.53828062474881842608722733893, 4.70539492821493616775028867546, 5.81982599204644179174743940537, 6.58598803678703886616545672542, 7.31916912698959727757925500846, 9.122219005705239874338202507098, 10.20601414122346704873503368121, 10.80055076306839713733597194110

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