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Properties

Label 2-507-13.12-c3-0-65
Degree $2$
Conductor $507$
Sign $-0.999 + 0.0304i$
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  − 2.82i·2-s + 3·3-s + 0.0423·4-s − 3.41i·5-s − 8.46i·6-s − 13.3i·7-s − 22.6i·8-s + 9·9-s − 9.62·10-s − 35.4i·11-s + 0.127·12-s − 37.6·14-s − 10.2i·15-s − 63.6·16-s − 69.6·17-s − 25.3i·18-s + ⋯
L(s)  = 1  − 0.997i·2-s + 0.577·3-s + 0.00529·4-s − 0.305i·5-s − 0.575i·6-s − 0.720i·7-s − 1.00i·8-s + 0.333·9-s − 0.304·10-s − 0.971i·11-s + 0.00305·12-s − 0.718·14-s − 0.176i·15-s − 0.994·16-s − 0.993·17-s − 0.332i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.999 + 0.0304i$
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: \((2,\ 507,\ (\ :3/2),\ -0.999 + 0.0304i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.292510525\)
\(L(\frac12)\) \(\approx\) \(2.292510525\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.82iT - 8T^{2} \)
5 \( 1 + 3.41iT - 125T^{2} \)
7 \( 1 + 13.3iT - 343T^{2} \)
11 \( 1 + 35.4iT - 1.33e3T^{2} \)
17 \( 1 + 69.6T + 4.91e3T^{2} \)
19 \( 1 - 12.4iT - 6.85e3T^{2} \)
23 \( 1 - 126.T + 1.21e4T^{2} \)
29 \( 1 + 179.T + 2.43e4T^{2} \)
31 \( 1 - 255. iT - 2.97e4T^{2} \)
37 \( 1 + 207. iT - 5.06e4T^{2} \)
41 \( 1 + 117. iT - 6.89e4T^{2} \)
43 \( 1 + 553.T + 7.95e4T^{2} \)
47 \( 1 + 62.9iT - 1.03e5T^{2} \)
53 \( 1 + 147.T + 1.48e5T^{2} \)
59 \( 1 + 274. iT - 2.05e5T^{2} \)
61 \( 1 - 603.T + 2.26e5T^{2} \)
67 \( 1 + 741. iT - 3.00e5T^{2} \)
71 \( 1 - 572. iT - 3.57e5T^{2} \)
73 \( 1 + 26.7iT - 3.89e5T^{2} \)
79 \( 1 + 207.T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.22e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.79e3iT - 9.12e5T^{2} \)
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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

−10.32561068004749733907836001635, −9.185334434550752307798244944156, −8.591156847977326100995156901875, −7.29229226848941976211585999273, −6.58051382335413724943722003512, −5.02184138373851721664120505837, −3.80231722733161273055320085336, −3.07123743162878647497247200627, −1.77403267576419551389516123453, −0.62006450937489735566237366481, 1.96443017962581208481191897702, 2.87297032797859163600815532861, 4.49956362432734426708236560561, 5.46311077613530904885961522839, 6.65229355533410503162021979993, 7.14504521041284720927530524295, 8.175092930952444345785812931479, 8.925681349286439993054261486213, 9.780075797336613721673365856159, 11.00631561645618172920006200340

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