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Properties

Label 2-507-39.20-c1-0-30
Degree $2$
Conductor $507$
Sign $0.466 + 0.884i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.5i)3-s + (1.73 − i)4-s + (2.86 + 0.767i)7-s + (−1.5 − 2.59i)9-s − 3.46i·12-s + (1.99 − 3.46i)16-s + (−2.09 + 7.83i)19-s + (3.63 − 3.63i)21-s + 5i·25-s − 5.19·27-s + (5.73 − 1.53i)28-s + (−7.83 − 7.83i)31-s + (−5.19 − 3i)36-s + (−0.562 − 2.09i)37-s + (1.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (0.866 − 0.5i)4-s + (1.08 + 0.290i)7-s + (−0.5 − 0.866i)9-s − 0.999i·12-s + (0.499 − 0.866i)16-s + (−0.481 + 1.79i)19-s + (0.792 − 0.792i)21-s + i·25-s − 1.00·27-s + (1.08 − 0.290i)28-s + (−1.40 − 1.40i)31-s + (−0.866 − 0.5i)36-s + (−0.0924 − 0.344i)37-s + (0.228 − 0.132i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.466 + 0.884i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.466 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84819 - 1.11517i\)
\(L(\frac12)\) \(\approx\) \(1.84819 - 1.11517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 1.5i)T \)
13 \( 1 \)
good2 \( 1 + (-1.73 + i)T^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 + (-2.86 - 0.767i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.09 - 7.83i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.83 + 7.83i)T + 31iT^{2} \)
37 \( 1 + (0.562 + 2.09i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.5 + 0.866i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.33 - 7.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.767 - 0.205i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (9.36 - 9.36i)T - 73iT^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.40 + 16.4i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.97805029075021376238185958940, −9.890016422493490846860637543265, −8.810177126042682205642821696353, −7.84666991913596832879564965631, −7.31928289946748772113047842652, −6.11197939753011840303256724559, −5.44782259795638836354684685465, −3.73642793903640894310048694360, −2.25594546526124639594162286500, −1.49099421647136330000285365921, 2.02110980973437629670988681889, 3.12054168560673643210740464439, 4.32240127111347398490904268138, 5.17789256277270035477392217238, 6.63857292807518547480113441025, 7.59545528227057060056025370739, 8.398376103433759317428709731094, 9.140620272523342520197468588336, 10.47427040590147717726679354057, 10.94955453179266433029681920133

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