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Properties

Label 2-507-13.10-c1-0-15
Degree $2$
Conductor $507$
Sign $0.964 - 0.265i$
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  + (1.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (1.5 + 0.866i)6-s + (3 + 1.73i)7-s + 1.73i·8-s + (−0.499 + 0.866i)9-s + (−3 + 1.73i)11-s + 12-s + 6·14-s + (2.49 + 4.33i)16-s + (3 − 5.19i)17-s + 1.73i·18-s + (−3 − 1.73i)19-s + 3.46i·21-s + (−3 + 5.19i)22-s + ⋯
L(s)  = 1  + (1.06 − 0.612i)2-s + (0.288 + 0.499i)3-s + (0.250 − 0.433i)4-s + (0.612 + 0.353i)6-s + (1.13 + 0.654i)7-s + 0.612i·8-s + (−0.166 + 0.288i)9-s + (−0.904 + 0.522i)11-s + 0.288·12-s + 1.60·14-s + (0.624 + 1.08i)16-s + (0.727 − 1.26i)17-s + 0.408i·18-s + (−0.688 − 0.397i)19-s + 0.755i·21-s + (−0.639 + 1.10i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.73192 + 0.368578i\)
\(L(\frac12)\) \(\approx\) \(2.73192 + 0.368578i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + (-3 - 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (-6 + 3.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6 - 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (9 + 5.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9 - 5.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 1.73i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12 - 6.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.22136490548050238700341296757, −10.32681603265505707390544823560, −9.222598148460125475631790069369, −8.269346925509371819170113405755, −7.50744271230282514883165145012, −5.74946229650646569566044859803, −4.95959133519996698833543753451, −4.38415144156080688565268646400, −2.94476769164125798464061857377, −2.16353542090953059532271659962, 1.39413030837277099293824169179, 3.19438602006445547739978125456, 4.29942459972492733596896472881, 5.25260540768434304842781425456, 6.11226896186732016946064861397, 7.17973129640586988754649034104, 7.937053758127701116969054986705, 8.707112751648232392472390546857, 10.28342073372220223343490437356, 10.81978271953389441699002435642

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