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Properties

Label 2-507-13.12-c3-0-40
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  − 4.22i·2-s − 3·3-s − 9.85·4-s − 5.85i·5-s + 12.6i·6-s + 24.1i·7-s + 7.85i·8-s + 9·9-s − 24.7·10-s + 33.8i·11-s + 29.5·12-s + 101.·14-s + 17.5i·15-s − 45.6·16-s + 49.3·17-s − 38.0i·18-s + ⋯
L(s)  = 1  − 1.49i·2-s − 0.577·3-s − 1.23·4-s − 0.524i·5-s + 0.862i·6-s + 1.30i·7-s + 0.347i·8-s + 0.333·9-s − 0.783·10-s + 0.928i·11-s + 0.711·12-s + 1.94·14-s + 0.302i·15-s − 0.713·16-s + 0.704·17-s − 0.498i·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

\(L(2)\) \(\approx\) \(1.361385491\)
\(L(\frac12)\) \(\approx\) \(1.361385491\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.22iT - 8T^{2} \)
5 \( 1 + 5.85iT - 125T^{2} \)
7 \( 1 - 24.1iT - 343T^{2} \)
11 \( 1 - 33.8iT - 1.33e3T^{2} \)
17 \( 1 - 49.3T + 4.91e3T^{2} \)
19 \( 1 + 76.8iT - 6.85e3T^{2} \)
23 \( 1 + 6.29T + 1.21e4T^{2} \)
29 \( 1 - 100.T + 2.43e4T^{2} \)
31 \( 1 + 307. iT - 2.97e4T^{2} \)
37 \( 1 - 76.0iT - 5.06e4T^{2} \)
41 \( 1 + 514. iT - 6.89e4T^{2} \)
43 \( 1 - 268.T + 7.95e4T^{2} \)
47 \( 1 - 460. iT - 1.03e5T^{2} \)
53 \( 1 - 67.8T + 1.48e5T^{2} \)
59 \( 1 + 25.2iT - 2.05e5T^{2} \)
61 \( 1 + 588.T + 2.26e5T^{2} \)
67 \( 1 + 1.00e3iT - 3.00e5T^{2} \)
71 \( 1 - 895. iT - 3.57e5T^{2} \)
73 \( 1 + 968. iT - 3.89e5T^{2} \)
79 \( 1 + 119.T + 4.93e5T^{2} \)
83 \( 1 - 480. iT - 5.71e5T^{2} \)
89 \( 1 + 1.08e3iT - 7.04e5T^{2} \)
97 \( 1 + 16.6iT - 9.12e5T^{2} \)
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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

−10.28779591101045225321249979592, −9.393087609835282213195942412346, −8.901223498808461405732354593463, −7.50478433272871216964318758094, −6.18419443726895411584425136935, −5.10121018583463566059219732400, −4.30673810389634406865984052269, −2.84067014431237230661366244718, −1.90802404368299994777597653302, −0.59132140320739781113644753036, 0.981693491481636556629495009090, 3.31389404118403436942887140501, 4.51301062447735934523132406122, 5.55282785305011076252829342194, 6.41035182610437586184043323154, 7.07911226129673112435065586566, 7.84386530226711883770577667684, 8.715917726808919628777279776749, 10.09996586529316291139306324629, 10.67929322533482222792785115889

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