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Properties

Label 2-507-507.131-c0-0-0
Degree $2$
Conductor $507$
Sign $0.806 - 0.590i$
Analytic cond. $0.253025$
Root an. cond. $0.503016$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.120 + 0.992i)3-s + (0.885 − 0.464i)4-s + (−0.180 + 0.159i)7-s + (−0.970 + 0.239i)9-s + (0.568 + 0.822i)12-s + 13-s + (0.568 − 0.822i)16-s − 1.94·19-s + (−0.180 − 0.159i)21-s + (−0.354 + 0.935i)25-s + (−0.354 − 0.935i)27-s + (−0.0854 + 0.225i)28-s + (−0.627 − 1.65i)31-s + (−0.748 + 0.663i)36-s + (0.530 + 1.39i)37-s + ⋯
L(s)  = 1  + (0.120 + 0.992i)3-s + (0.885 − 0.464i)4-s + (−0.180 + 0.159i)7-s + (−0.970 + 0.239i)9-s + (0.568 + 0.822i)12-s + 13-s + (0.568 − 0.822i)16-s − 1.94·19-s + (−0.180 − 0.159i)21-s + (−0.354 + 0.935i)25-s + (−0.354 − 0.935i)27-s + (−0.0854 + 0.225i)28-s + (−0.627 − 1.65i)31-s + (−0.748 + 0.663i)36-s + (0.530 + 1.39i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.806 - 0.590i$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ 0.806 - 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.043743021\)
\(L(\frac12)\) \(\approx\) \(1.043743021\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.120 - 0.992i)T \)
13 \( 1 - T \)
good2 \( 1 + (-0.885 + 0.464i)T^{2} \)
5 \( 1 + (0.354 - 0.935i)T^{2} \)
7 \( 1 + (0.180 - 0.159i)T + (0.120 - 0.992i)T^{2} \)
11 \( 1 + (-0.885 - 0.464i)T^{2} \)
17 \( 1 + (-0.120 + 0.992i)T^{2} \)
19 \( 1 + 1.94T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.885 + 0.464i)T^{2} \)
31 \( 1 + (0.627 + 1.65i)T + (-0.748 + 0.663i)T^{2} \)
37 \( 1 + (-0.530 - 1.39i)T + (-0.748 + 0.663i)T^{2} \)
41 \( 1 + (0.970 - 0.239i)T^{2} \)
43 \( 1 + (-0.530 + 1.39i)T + (-0.748 - 0.663i)T^{2} \)
47 \( 1 + (-0.568 - 0.822i)T^{2} \)
53 \( 1 + (-0.120 + 0.992i)T^{2} \)
59 \( 1 + (0.354 - 0.935i)T^{2} \)
61 \( 1 + (1.32 + 1.17i)T + (0.120 + 0.992i)T^{2} \)
67 \( 1 + (0.627 + 0.329i)T + (0.568 + 0.822i)T^{2} \)
71 \( 1 + (0.970 - 0.239i)T^{2} \)
73 \( 1 + (1.10 + 0.271i)T + (0.885 + 0.464i)T^{2} \)
79 \( 1 + (-0.213 - 0.112i)T + (0.568 + 0.822i)T^{2} \)
83 \( 1 + (0.970 + 0.239i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.13 + 1.64i)T + (-0.354 - 0.935i)T^{2} \)
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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

−11.01161050751841250293800568786, −10.49772023441173479507436869239, −9.539377620579378409183663370791, −8.711142299328751114411223960387, −7.68083202984683750479827736526, −6.30748362377606855223624088807, −5.80940327356968749208548626921, −4.48368264033672289791408752688, −3.37949326430949326603062545325, −2.09987512377027086902394265016, 1.73013882416810019294517259231, 2.85670948357188494551015663799, 4.06239127593891400489189079797, 5.92993631640439693871875434226, 6.49144299309225133249445250382, 7.34767911279986896026955672839, 8.275249664538902716456163107870, 8.888624309055078522451229209575, 10.52533898451594106783145202185, 11.05021872200800012082841948694

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