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Properties

Label 2-507-13.12-c3-0-4
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  + 4.74i·2-s − 3·3-s − 14.4·4-s + 4.51i·5-s − 14.2i·6-s + 7.48i·7-s − 30.7i·8-s + 9·9-s − 21.4·10-s + 66.8i·11-s + 43.4·12-s − 35.4·14-s − 13.5i·15-s + 29.8·16-s − 96.9·17-s + 42.6i·18-s + ⋯
L(s)  = 1  + 1.67i·2-s − 0.577·3-s − 1.81·4-s + 0.403i·5-s − 0.967i·6-s + 0.404i·7-s − 1.35i·8-s + 0.333·9-s − 0.677·10-s + 1.83i·11-s + 1.04·12-s − 0.677·14-s − 0.233i·15-s + 0.467·16-s − 1.38·17-s + 0.558i·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

\(L(2)\) \(\approx\) \(0.4396490062\)
\(L(\frac12)\) \(\approx\) \(0.4396490062\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.74iT - 8T^{2} \)
5 \( 1 - 4.51iT - 125T^{2} \)
7 \( 1 - 7.48iT - 343T^{2} \)
11 \( 1 - 66.8iT - 1.33e3T^{2} \)
17 \( 1 + 96.9T + 4.91e3T^{2} \)
19 \( 1 - 31.4iT - 6.85e3T^{2} \)
23 \( 1 + 183.T + 1.21e4T^{2} \)
29 \( 1 - 112.T + 2.43e4T^{2} \)
31 \( 1 + 77.2iT - 2.97e4T^{2} \)
37 \( 1 + 54.7iT - 5.06e4T^{2} \)
41 \( 1 - 451. iT - 6.89e4T^{2} \)
43 \( 1 - 113.T + 7.95e4T^{2} \)
47 \( 1 - 42.2iT - 1.03e5T^{2} \)
53 \( 1 + 530.T + 1.48e5T^{2} \)
59 \( 1 + 219. iT - 2.05e5T^{2} \)
61 \( 1 - 822.T + 2.26e5T^{2} \)
67 \( 1 + 872. iT - 3.00e5T^{2} \)
71 \( 1 + 100. iT - 3.57e5T^{2} \)
73 \( 1 - 165. iT - 3.89e5T^{2} \)
79 \( 1 + 545.T + 4.93e5T^{2} \)
83 \( 1 + 454. iT - 5.71e5T^{2} \)
89 \( 1 - 230. iT - 7.04e5T^{2} \)
97 \( 1 + 1.08e3iT - 9.12e5T^{2} \)
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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

−11.30311234207549476154214539788, −10.11687619136694012326268722860, −9.411465900557261202625276492112, −8.313278708315992248812279083722, −7.44922095453974837189074114046, −6.65311217567793535573229781210, −6.08865769098392054442791668576, −4.88343881522857811485405839774, −4.28911319253247541236315579394, −2.15168742002513748764091430035, 0.17523120671042420412820680011, 1.02403397866628396325199985424, 2.44500005329015641116888291515, 3.68257831788277959159050678930, 4.50838069925578808164581709016, 5.65903922332425782203300470214, 6.79240861413572123349156748972, 8.428637761210364309241851701835, 8.936138762029080289561322610554, 10.10952267286214454097517223007

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