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Properties

Label 2-495-3.2-c2-0-19
Degree $2$
Conductor $495$
Sign $0.577 + 0.816i$
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  + 2.92i·2-s − 4.55·4-s − 2.23i·5-s − 2.18·7-s − 1.63i·8-s + 6.54·10-s − 3.31i·11-s − 9.32·13-s − 6.38i·14-s − 13.4·16-s − 6.11i·17-s − 28.8·19-s + 10.1i·20-s + 9.70·22-s − 31.8i·23-s + ⋯
L(s)  = 1  + 1.46i·2-s − 1.13·4-s − 0.447i·5-s − 0.312·7-s − 0.204i·8-s + 0.654·10-s − 0.301i·11-s − 0.717·13-s − 0.456i·14-s − 0.840·16-s − 0.359i·17-s − 1.51·19-s + 0.509i·20-s + 0.441·22-s − 1.38i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (386, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4073511756\)
\(L(\frac12)\) \(\approx\) \(0.4073511756\)
\(L(2)\) 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 \( 1 + 2.23iT \)
11 \( 1 + 3.31iT \)
good2 \( 1 - 2.92iT - 4T^{2} \)
7 \( 1 + 2.18T + 49T^{2} \)
13 \( 1 + 9.32T + 169T^{2} \)
17 \( 1 + 6.11iT - 289T^{2} \)
19 \( 1 + 28.8T + 361T^{2} \)
23 \( 1 + 31.8iT - 529T^{2} \)
29 \( 1 + 16.1iT - 841T^{2} \)
31 \( 1 - 14.9T + 961T^{2} \)
37 \( 1 - 21.4T + 1.36e3T^{2} \)
41 \( 1 + 31.6iT - 1.68e3T^{2} \)
43 \( 1 + 34.3T + 1.84e3T^{2} \)
47 \( 1 - 42.2iT - 2.20e3T^{2} \)
53 \( 1 + 51.1iT - 2.80e3T^{2} \)
59 \( 1 - 22.2iT - 3.48e3T^{2} \)
61 \( 1 + 67.3T + 3.72e3T^{2} \)
67 \( 1 - 107.T + 4.48e3T^{2} \)
71 \( 1 - 21.6iT - 5.04e3T^{2} \)
73 \( 1 - 44.1T + 5.32e3T^{2} \)
79 \( 1 + 157.T + 6.24e3T^{2} \)
83 \( 1 + 95.6iT - 6.88e3T^{2} \)
89 \( 1 - 98.6iT - 7.92e3T^{2} \)
97 \( 1 - 35.1T + 9.40e3T^{2} \)
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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

−10.41766193013506027212378810952, −9.378305054473598144950253107536, −8.522224826919408205913572909743, −7.895790944957750858235381993076, −6.76242777684887722716266737635, −6.19175966262179152761493350142, −5.06623598713517445200461650619, −4.26682358540648943284948469038, −2.44876049145853732481190738960, −0.15458672077625544275430043598, 1.67831030169556038163096506446, 2.74447323580443396654158454698, 3.74970460079506258348712320153, 4.78919089835429499484954789298, 6.26544714496001023213058921564, 7.21661379905818267150057038120, 8.451819383064372948077983679501, 9.575872305410891878879709911456, 10.07299338473499555942037671138, 10.93763120242280047779756031317

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