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Properties

Label 2-507-13.12-c3-0-70
Degree $2$
Conductor $507$
Sign $-0.246 - 0.969i$
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  − 4.03i·2-s + 3·3-s − 8.24·4-s − 8.08i·5-s − 12.0i·6-s + 5.95i·7-s + 0.978i·8-s + 9·9-s − 32.5·10-s + 17.2i·11-s − 24.7·12-s + 23.9·14-s − 24.2i·15-s − 61.9·16-s − 92.9·17-s − 36.2i·18-s + ⋯
L(s)  = 1  − 1.42i·2-s + 0.577·3-s − 1.03·4-s − 0.723i·5-s − 0.822i·6-s + 0.321i·7-s + 0.0432i·8-s + 0.333·9-s − 1.03·10-s + 0.472i·11-s − 0.594·12-s + 0.457·14-s − 0.417i·15-s − 0.968·16-s − 1.32·17-s − 0.474i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.246 - 0.969i$
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: \((2,\ 507,\ (\ :3/2),\ -0.246 - 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9102365173\)
\(L(\frac12)\) \(\approx\) \(0.9102365173\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.03iT - 8T^{2} \)
5 \( 1 + 8.08iT - 125T^{2} \)
7 \( 1 - 5.95iT - 343T^{2} \)
11 \( 1 - 17.2iT - 1.33e3T^{2} \)
17 \( 1 + 92.9T + 4.91e3T^{2} \)
19 \( 1 + 13.3iT - 6.85e3T^{2} \)
23 \( 1 + 219.T + 1.21e4T^{2} \)
29 \( 1 + 199.T + 2.43e4T^{2} \)
31 \( 1 + 307. iT - 2.97e4T^{2} \)
37 \( 1 + 333. iT - 5.06e4T^{2} \)
41 \( 1 - 200. iT - 6.89e4T^{2} \)
43 \( 1 + 116.T + 7.95e4T^{2} \)
47 \( 1 - 338. iT - 1.03e5T^{2} \)
53 \( 1 + 26.6T + 1.48e5T^{2} \)
59 \( 1 + 280. iT - 2.05e5T^{2} \)
61 \( 1 + 207.T + 2.26e5T^{2} \)
67 \( 1 - 285. iT - 3.00e5T^{2} \)
71 \( 1 - 317. iT - 3.57e5T^{2} \)
73 \( 1 + 63.0iT - 3.89e5T^{2} \)
79 \( 1 + 623.T + 4.93e5T^{2} \)
83 \( 1 + 659. iT - 5.71e5T^{2} \)
89 \( 1 - 1.27e3iT - 7.04e5T^{2} \)
97 \( 1 + 603. iT - 9.12e5T^{2} \)
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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

−9.765731619483719180259770688801, −9.296373206116021938287760069643, −8.445259078695397168314086374326, −7.34594004052938891460070787930, −5.98474075389011279670404773967, −4.51722003690120957773331696550, −3.90898540915292892814031018491, −2.46258784179629657308412769628, −1.79636108770879322512703461636, −0.23645328594450249087798782175, 2.10527072885894620071410903060, 3.50140723367906235502223883692, 4.65572932905773464264984823042, 5.89089963542773711230466307083, 6.73096271106071718934325580313, 7.34813637670411524710483909253, 8.333888419794854095782881319265, 8.903927019465329588243169818219, 10.13350616521829938239375461883, 10.93945461779804443544167396922

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