Properties

Label 2-507-507.350-c0-0-0
Degree $2$
Conductor $507$
Sign $-0.230 + 0.973i$
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  + (−0.885 − 0.464i)3-s + (0.354 − 0.935i)4-s + (−0.222 − 0.902i)7-s + (0.568 + 0.822i)9-s + (−0.748 + 0.663i)12-s − 13-s + (−0.748 − 0.663i)16-s − 1.64i·19-s + (−0.222 + 0.902i)21-s + (−0.120 + 0.992i)25-s + (−0.120 − 0.992i)27-s + (−0.922 − 0.112i)28-s + (1.85 − 0.225i)31-s + (0.970 − 0.239i)36-s + (0.475 − 0.0576i)37-s + ⋯
L(s)  = 1  + (−0.885 − 0.464i)3-s + (0.354 − 0.935i)4-s + (−0.222 − 0.902i)7-s + (0.568 + 0.822i)9-s + (−0.748 + 0.663i)12-s − 13-s + (−0.748 − 0.663i)16-s − 1.64i·19-s + (−0.222 + 0.902i)21-s + (−0.120 + 0.992i)25-s + (−0.120 − 0.992i)27-s + (−0.922 − 0.112i)28-s + (1.85 − 0.225i)31-s + (0.970 − 0.239i)36-s + (0.475 − 0.0576i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.230 + 0.973i$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (350, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ -0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6600435850\)
\(L(\frac12)\) \(\approx\) \(0.6600435850\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.885 + 0.464i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.354 + 0.935i)T^{2} \)
5 \( 1 + (0.120 - 0.992i)T^{2} \)
7 \( 1 + (0.222 + 0.902i)T + (-0.885 + 0.464i)T^{2} \)
11 \( 1 + (-0.354 - 0.935i)T^{2} \)
17 \( 1 + (-0.885 + 0.464i)T^{2} \)
19 \( 1 + 1.64iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.354 - 0.935i)T^{2} \)
31 \( 1 + (-1.85 + 0.225i)T + (0.970 - 0.239i)T^{2} \)
37 \( 1 + (-0.475 + 0.0576i)T + (0.970 - 0.239i)T^{2} \)
41 \( 1 + (0.568 + 0.822i)T^{2} \)
43 \( 1 + (0.234 - 1.92i)T + (-0.970 - 0.239i)T^{2} \)
47 \( 1 + (-0.748 + 0.663i)T^{2} \)
53 \( 1 + (-0.885 + 0.464i)T^{2} \)
59 \( 1 + (0.120 - 0.992i)T^{2} \)
61 \( 1 + (0.688 + 0.169i)T + (0.885 + 0.464i)T^{2} \)
67 \( 1 + (-1.85 + 0.704i)T + (0.748 - 0.663i)T^{2} \)
71 \( 1 + (0.568 + 0.822i)T^{2} \)
73 \( 1 + (-1.09 - 0.753i)T + (0.354 + 0.935i)T^{2} \)
79 \( 1 + (-0.627 - 1.65i)T + (-0.748 + 0.663i)T^{2} \)
83 \( 1 + (0.568 - 0.822i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.120 - 0.992i)T^{2} \)
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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

−11.06585442723797117773600949542, −10.05113585090467166788024401921, −9.513550804175378027927992805839, −7.84591671227510897167458674935, −6.94601561283530896843376856936, −6.44741343413880145472575813346, −5.22044525972759150737143927980, −4.53044390048567912855944843545, −2.53655730238305147466752123619, −0.944782389261747953360958677341, 2.37411744666729249688853089711, 3.66679955168328498820081888320, 4.75655647608536940554894542405, 5.88669455472781977766376540017, 6.66635836399058625110976275422, 7.78672305889136123586425109153, 8.691247056268792387070921434759, 9.789218540793685901331312757033, 10.46482042145324701060700157143, 11.64516869939598764650639713955

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