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Properties

Label 2-507-13.12-c3-0-1
Degree $2$
Conductor $507$
Sign $-0.832 + 0.554i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.33i·2-s − 3·3-s − 20.4·4-s − 16.4i·5-s − 15.9i·6-s − 9.67i·7-s − 66.1i·8-s + 9·9-s + 87.4·10-s − 27.5i·11-s + 61.2·12-s + 51.5·14-s + 49.2i·15-s + 189.·16-s − 107.·17-s + 47.9i·18-s + ⋯
L(s)  = 1  + 1.88i·2-s − 0.577·3-s − 2.55·4-s − 1.46i·5-s − 1.08i·6-s − 0.522i·7-s − 2.92i·8-s + 0.333·9-s + 2.76·10-s − 0.756i·11-s + 1.47·12-s + 0.984·14-s + 0.847i·15-s + 2.95·16-s − 1.53·17-s + 0.628i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1702981080\)
\(L(\frac12)\) \(\approx\) \(0.1702981080\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 \)
good2 \( 1 - 5.33iT - 8T^{2} \)
5 \( 1 + 16.4iT - 125T^{2} \)
7 \( 1 + 9.67iT - 343T^{2} \)
11 \( 1 + 27.5iT - 1.33e3T^{2} \)
17 \( 1 + 107.T + 4.91e3T^{2} \)
19 \( 1 + 2.24iT - 6.85e3T^{2} \)
23 \( 1 + 41.8T + 1.21e4T^{2} \)
29 \( 1 - 61.6T + 2.43e4T^{2} \)
31 \( 1 - 191. iT - 2.97e4T^{2} \)
37 \( 1 + 98.4iT - 5.06e4T^{2} \)
41 \( 1 + 30.7iT - 6.89e4T^{2} \)
43 \( 1 + 238.T + 7.95e4T^{2} \)
47 \( 1 - 511. iT - 1.03e5T^{2} \)
53 \( 1 - 492.T + 1.48e5T^{2} \)
59 \( 1 + 484. iT - 2.05e5T^{2} \)
61 \( 1 + 444.T + 2.26e5T^{2} \)
67 \( 1 - 190. iT - 3.00e5T^{2} \)
71 \( 1 - 484. iT - 3.57e5T^{2} \)
73 \( 1 - 957. iT - 3.89e5T^{2} \)
79 \( 1 + 375.T + 4.93e5T^{2} \)
83 \( 1 + 715. iT - 5.71e5T^{2} \)
89 \( 1 - 1.03e3iT - 7.04e5T^{2} \)
97 \( 1 - 65.5iT - 9.12e5T^{2} \)
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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

−10.91028655301023965388577866695, −9.706334558023056170382946291130, −8.774923041839275523980115450585, −8.387359070152541623797841371777, −7.26398602686369633889805688011, −6.41057129125528092480748131448, −5.53610999143646727010340213709, −4.74645235055810286344973248648, −4.06289993272182890002597716842, −0.971807518913808236602584366253, 0.07239028050315168898227894509, 1.95210767810578003530375145422, 2.64764352763160945615163174273, 3.84704005893333192527995394339, 4.79485703398272344301402283011, 6.10844369153206273409564213635, 7.19453914091496924508515000219, 8.589187474142925575930437354279, 9.582837421103639044288827169397, 10.34512076140767421402505085884

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