Properties

Label 2-507-13.12-c1-0-20
Degree $2$
Conductor $507$
Sign $-0.691 + 0.722i$
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  − 1.80i·2-s + 3-s − 1.24·4-s − 1.44i·5-s − 1.80i·6-s + 3.44i·7-s − 1.35i·8-s + 9-s − 2.60·10-s − 5.18i·11-s − 1.24·12-s + 6.20·14-s − 1.44i·15-s − 4.93·16-s + 0.753·17-s − 1.80i·18-s + ⋯
L(s)  = 1  − 1.27i·2-s + 0.577·3-s − 0.623·4-s − 0.646i·5-s − 0.735i·6-s + 1.30i·7-s − 0.479i·8-s + 0.333·9-s − 0.823·10-s − 1.56i·11-s − 0.359·12-s + 1.65·14-s − 0.373i·15-s − 1.23·16-s + 0.182·17-s − 0.424i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.690366 - 1.61716i\)
\(L(\frac12)\) \(\approx\) \(0.690366 - 1.61716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
13 \( 1 \)
good2 \( 1 + 1.80iT - 2T^{2} \)
5 \( 1 + 1.44iT - 5T^{2} \)
7 \( 1 - 3.44iT - 7T^{2} \)
11 \( 1 + 5.18iT - 11T^{2} \)
17 \( 1 - 0.753T + 17T^{2} \)
19 \( 1 + 7.96iT - 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 3.91T + 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 - 6.24iT - 37T^{2} \)
41 \( 1 + 1.80iT - 41T^{2} \)
43 \( 1 - 7.09T + 43T^{2} \)
47 \( 1 - 10.5iT - 47T^{2} \)
53 \( 1 + 3.08T + 53T^{2} \)
59 \( 1 - 1.87iT - 59T^{2} \)
61 \( 1 - 3.34T + 61T^{2} \)
67 \( 1 - 4.54iT - 67T^{2} \)
71 \( 1 - 9.11iT - 71T^{2} \)
73 \( 1 - 2.95iT - 73T^{2} \)
79 \( 1 + 9.43T + 79T^{2} \)
83 \( 1 - 6.46iT - 83T^{2} \)
89 \( 1 - 1.15iT - 89T^{2} \)
97 \( 1 + 8.65iT - 97T^{2} \)
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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

−10.89570649854550246496585626676, −9.553910303922076851578856017307, −8.918776482829683718927052631117, −8.486664999424894007055833509166, −6.94586781274760156417771851723, −5.69319865909817833229851154249, −4.60814935532054668690156607025, −3.16978837543777033337460658369, −2.60781302073818172050014928076, −1.06490786971069116226107588199, 2.01890488757516149850030633341, 3.67340077188047881825447126990, 4.63888975635074699043875745028, 5.94168052563671430389541916584, 7.10503109826280366554765691167, 7.33181856132040904951112933473, 8.121825428960928016400926081703, 9.394130547054549034663465111676, 10.21498434275851988567688332437, 10.96872732954285040827843857846

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