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Properties

Label 2-495-5.2-c2-0-47
Degree $2$
Conductor $495$
Sign $-0.642 - 0.766i$
Analytic cond. $13.4877$
Root an. cond. $3.67257$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.762 − 0.762i)2-s − 2.83i·4-s + (4.94 − 0.716i)5-s + (−5.64 − 5.64i)7-s + (−5.21 + 5.21i)8-s + (−4.32 − 3.22i)10-s − 3.31·11-s + (−13.3 + 13.3i)13-s + 8.61i·14-s − 3.38·16-s + (−21.6 − 21.6i)17-s + 34.8i·19-s + (−2.03 − 14.0i)20-s + (2.53 + 2.53i)22-s + (15.6 − 15.6i)23-s + ⋯
L(s)  = 1  + (−0.381 − 0.381i)2-s − 0.708i·4-s + (0.989 − 0.143i)5-s + (−0.806 − 0.806i)7-s + (−0.651 + 0.651i)8-s + (−0.432 − 0.322i)10-s − 0.301·11-s + (−1.02 + 1.02i)13-s + 0.615i·14-s − 0.211·16-s + (−1.27 − 1.27i)17-s + 1.83i·19-s + (−0.101 − 0.701i)20-s + (0.115 + 0.115i)22-s + (0.679 − 0.679i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2097669205\)
\(L(\frac12)\) \(\approx\) \(0.2097669205\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-4.94 + 0.716i)T \)
11 \( 1 + 3.31T \)
good2 \( 1 + (0.762 + 0.762i)T + 4iT^{2} \)
7 \( 1 + (5.64 + 5.64i)T + 49iT^{2} \)
13 \( 1 + (13.3 - 13.3i)T - 169iT^{2} \)
17 \( 1 + (21.6 + 21.6i)T + 289iT^{2} \)
19 \( 1 - 34.8iT - 361T^{2} \)
23 \( 1 + (-15.6 + 15.6i)T - 529iT^{2} \)
29 \( 1 - 11.8iT - 841T^{2} \)
31 \( 1 + 24.8T + 961T^{2} \)
37 \( 1 + (9.97 + 9.97i)T + 1.36e3iT^{2} \)
41 \( 1 - 13.0T + 1.68e3T^{2} \)
43 \( 1 + (2.85 - 2.85i)T - 1.84e3iT^{2} \)
47 \( 1 + (21.6 + 21.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (-7.24 + 7.24i)T - 2.80e3iT^{2} \)
59 \( 1 - 57.4iT - 3.48e3T^{2} \)
61 \( 1 + 52.6T + 3.72e3T^{2} \)
67 \( 1 + (35.3 + 35.3i)T + 4.48e3iT^{2} \)
71 \( 1 + 94.5T + 5.04e3T^{2} \)
73 \( 1 + (72.5 - 72.5i)T - 5.32e3iT^{2} \)
79 \( 1 - 84.9iT - 6.24e3T^{2} \)
83 \( 1 + (-68.0 + 68.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 22.9iT - 7.92e3T^{2} \)
97 \( 1 + (27.4 + 27.4i)T + 9.40e3iT^{2} \)
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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

−10.13382922917764183194675349398, −9.447633817174262103126412593672, −8.853700937412280417093794658096, −7.19607674079945618756638519824, −6.51354799825230015946267254100, −5.46047346355407578993528574236, −4.45736495328375633873517122252, −2.73768596682308653598022721595, −1.66776117053886560356637937888, −0.087442221886681966625961801328, 2.39557759582252830491915355472, 3.12977749527826628923975888153, 4.83596504323567763277466801735, 5.96570090436472464465693887373, 6.73534912732203400107993814902, 7.64129308296603062979351173866, 8.927296597600502706314997819041, 9.189612183984158127184643595984, 10.23072129038662953393558669174, 11.19503233467770212363169706686

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