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Properties

Label 2-507-39.2-c1-0-1
Degree $2$
Conductor $507$
Sign $0.106 + 0.994i$
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  + (−0.339 + 1.26i)2-s + (−0.228 + 1.71i)3-s + (0.239 + 0.138i)4-s + (−2.12 − 2.12i)5-s + (−2.09 − 0.873i)6-s + (−2.82 + 0.755i)7-s + (−2.11 + 2.11i)8-s + (−2.89 − 0.785i)9-s + (3.41 − 1.97i)10-s + (2.57 + 0.689i)11-s + (−0.291 + 0.379i)12-s − 3.83i·14-s + (4.13 − 3.16i)15-s + (−1.68 − 2.91i)16-s + (0.0994 − 0.172i)17-s + (1.98 − 3.40i)18-s + ⋯
L(s)  = 1  + (−0.240 + 0.896i)2-s + (−0.132 + 0.991i)3-s + (0.119 + 0.0690i)4-s + (−0.950 − 0.950i)5-s + (−0.857 − 0.356i)6-s + (−1.06 + 0.285i)7-s + (−0.747 + 0.747i)8-s + (−0.965 − 0.261i)9-s + (1.08 − 0.623i)10-s + (0.775 + 0.207i)11-s + (−0.0842 + 0.109i)12-s − 1.02i·14-s + (1.06 − 0.816i)15-s + (−0.421 − 0.729i)16-s + (0.0241 − 0.0417i)17-s + (0.466 − 0.802i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.106 + 0.994i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.106 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0611328 - 0.0549273i\)
\(L(\frac12)\) \(\approx\) \(0.0611328 - 0.0549273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.228 - 1.71i)T \)
13 \( 1 \)
good2 \( 1 + (0.339 - 1.26i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (2.12 + 2.12i)T + 5iT^{2} \)
7 \( 1 + (2.82 - 0.755i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.57 - 0.689i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.0994 + 0.172i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.37 + 5.12i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.65 + 2.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.23 - 1.86i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.550 + 0.550i)T - 31iT^{2} \)
37 \( 1 + (1.32 - 4.92i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.986 + 3.68i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (10.0 + 5.82i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.29 - 4.29i)T - 47iT^{2} \)
53 \( 1 - 13.6iT - 53T^{2} \)
59 \( 1 + (-0.864 - 3.22i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.35 + 2.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.28 + 1.95i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.16 - 1.38i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.46 - 3.46i)T + 73iT^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + (-1.84 - 1.84i)T + 83iT^{2} \)
89 \( 1 + (1.06 + 0.284i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.14 - 11.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.77313554587332323028703661136, −10.65781427362836003402611239801, −9.393457161377462196821534747785, −8.921277614661150813578483237361, −8.180741330465602060411656096273, −6.96873930771999595149790835437, −6.16444472798502814128202520733, −5.05250346434389197019332665007, −4.07745034528533778201572644147, −2.95587886587010259666561894957, 0.05072505562748990563041366991, 1.75115467572118682339971477270, 3.18425034048490898041298251352, 3.68749186172174685394880844088, 5.98742121613178838002619612515, 6.62869357256285387634747203071, 7.33100719225907860629268366341, 8.385384926776328174559652336410, 9.651683970929627109648893137331, 10.37580504629686831253312915040

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