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Properties

Label 2-507-13.12-c3-0-10
Degree $2$
Conductor $507$
Sign $-0.277 - 0.960i$
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  − 3·3-s + 8·4-s + 5.19i·5-s − 10.3i·7-s + 9·9-s + 51.9i·11-s − 24·12-s − 15.5i·15-s + 64·16-s − 117·17-s + 24.2i·19-s + 41.5i·20-s + 31.1i·21-s + 18·23-s + 98·25-s + ⋯
L(s)  = 1  − 0.577·3-s + 4-s + 0.464i·5-s − 0.561i·7-s + 0.333·9-s + 1.42i·11-s − 0.577·12-s − 0.268i·15-s + 16-s − 1.66·17-s + 0.292i·19-s + 0.464i·20-s + 0.323i·21-s + 0.163·23-s + 0.784·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.277 - 0.960i$
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: \((2,\ 507,\ (\ :3/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.450193101\)
\(L(\frac12)\) \(\approx\) \(1.450193101\)
\(L(\frac{5}{2})\) 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 \( 1 - 8T^{2} \)
5 \( 1 - 5.19iT - 125T^{2} \)
7 \( 1 + 10.3iT - 343T^{2} \)
11 \( 1 - 51.9iT - 1.33e3T^{2} \)
17 \( 1 + 117T + 4.91e3T^{2} \)
19 \( 1 - 24.2iT - 6.85e3T^{2} \)
23 \( 1 - 18T + 1.21e4T^{2} \)
29 \( 1 + 99T + 2.43e4T^{2} \)
31 \( 1 - 193. iT - 2.97e4T^{2} \)
37 \( 1 - 112. iT - 5.06e4T^{2} \)
41 \( 1 + 36.3iT - 6.89e4T^{2} \)
43 \( 1 + 82T + 7.95e4T^{2} \)
47 \( 1 + 72.7iT - 1.03e5T^{2} \)
53 \( 1 + 261T + 1.48e5T^{2} \)
59 \( 1 - 789. iT - 2.05e5T^{2} \)
61 \( 1 + 719T + 2.26e5T^{2} \)
67 \( 1 + 703. iT - 3.00e5T^{2} \)
71 \( 1 - 467. iT - 3.57e5T^{2} \)
73 \( 1 - 684. iT - 3.89e5T^{2} \)
79 \( 1 + 440T + 4.93e5T^{2} \)
83 \( 1 - 1.19e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.51e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.15e3iT - 9.12e5T^{2} \)
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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

−10.68146975157105867078297431534, −10.30434883002948537473483227863, −9.110307475348683644266951629779, −7.69352686227346422618522077820, −6.89568623477962945494789150529, −6.53130323875136454742302039664, −5.13397355008655610771556281911, −4.07515370850593451866546208597, −2.61907955212781802327804428360, −1.49165933183868599628163184016, 0.45477283372033126823421343651, 1.95192339247304154875575208710, 3.16696449929894139800470380346, 4.63445663751805936925757277499, 5.79927056157897466393676768853, 6.33221169210607497419078390249, 7.39201319710844994345600880938, 8.523804449304950013507301119182, 9.232427400492467480500414024920, 10.59435348400183086916922527739

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