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Properties

Label 2-507-507.431-c1-0-40
Degree $2$
Conductor $507$
Sign $0.965 + 0.261i$
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  + (1.19 + 1.24i)3-s + (1.80 − 0.857i)4-s + (−0.697 − 4.91i)7-s + (−0.120 + 2.99i)9-s + (3.23 + 1.22i)12-s + (2.59 − 2.5i)13-s + (2.52 − 3.09i)16-s + (0.239 + 0.895i)19-s + (5.30 − 6.76i)21-s + (−3.31 + 3.74i)25-s + (−3.88 + 3.44i)27-s + (−5.47 − 8.28i)28-s + (−0.590 + 9.76i)31-s + (2.35 + 5.51i)36-s + (−1.60 − 3.21i)37-s + ⋯
L(s)  = 1  + (0.692 + 0.721i)3-s + (0.903 − 0.428i)4-s + (−0.263 − 1.85i)7-s + (−0.0402 + 0.999i)9-s + (0.935 + 0.354i)12-s + (0.720 − 0.693i)13-s + (0.632 − 0.774i)16-s + (0.0550 + 0.205i)19-s + (1.15 − 1.47i)21-s + (−0.663 + 0.748i)25-s + (−0.748 + 0.663i)27-s + (−1.03 − 1.56i)28-s + (−0.106 + 1.75i)31-s + (0.391 + 0.919i)36-s + (−0.264 − 0.529i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.965 + 0.261i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.965 + 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11274 - 0.281202i\)
\(L(\frac12)\) \(\approx\) \(2.11274 - 0.281202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.19 - 1.24i)T \)
13 \( 1 + (-2.59 + 2.5i)T \)
good2 \( 1 + (-1.80 + 0.857i)T^{2} \)
5 \( 1 + (3.31 - 3.74i)T^{2} \)
7 \( 1 + (0.697 + 4.91i)T + (-6.72 + 1.94i)T^{2} \)
11 \( 1 + (0.885 + 10.9i)T^{2} \)
17 \( 1 + (4.72 + 16.3i)T^{2} \)
19 \( 1 + (-0.239 - 0.895i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-12.4 - 26.2i)T^{2} \)
31 \( 1 + (0.590 - 9.76i)T + (-30.7 - 3.73i)T^{2} \)
37 \( 1 + (1.60 + 3.21i)T + (-22.2 + 29.5i)T^{2} \)
41 \( 1 + (-40.9 - 1.65i)T^{2} \)
43 \( 1 + (-4.15 - 12.4i)T + (-34.3 + 25.8i)T^{2} \)
47 \( 1 + (43.9 + 16.6i)T^{2} \)
53 \( 1 + (51.4 + 12.6i)T^{2} \)
59 \( 1 + (57.8 + 11.8i)T^{2} \)
61 \( 1 + (-5.31 - 2.26i)T + (42.2 + 43.9i)T^{2} \)
67 \( 1 + (3.93 + 11.0i)T + (-51.8 + 42.3i)T^{2} \)
71 \( 1 + (37.9 - 60.0i)T^{2} \)
73 \( 1 + (2.54 + 8.16i)T + (-60.0 + 41.4i)T^{2} \)
79 \( 1 + (-4.91 - 7.12i)T + (-28.0 + 73.8i)T^{2} \)
83 \( 1 + (-38.5 + 73.4i)T^{2} \)
89 \( 1 + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.06 - 5.80i)T + (30.7 - 92.0i)T^{2} \)
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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

−10.66124739972017150798162684647, −10.22814459694691609916349008216, −9.322165459100384000022683256802, −7.980272443073550183842859587731, −7.38703709227506716003173745736, −6.37124249867674468043793616194, −5.11254743197548808652694262116, −3.86457352689634274560364017477, −3.10093851687025231916355070333, −1.37348716144680756195462803673, 1.97852635783212980782300886800, 2.62518628593211887714456481179, 3.81279538992979488782366083133, 5.78107517122226847659805922208, 6.33960337674344375248077244052, 7.34021601684211415190694627658, 8.383390270705275246117529063688, 8.860307512079253073756389031353, 9.865620312476494513081416354719, 11.39484783024688752230548981219

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