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Properties

Label 2-495-55.43-c1-0-4
Degree $2$
Conductor $495$
Sign $0.624 - 0.781i$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
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  + (−0.478 − 0.478i)2-s − 1.54i·4-s + (1.04 + 1.97i)5-s + (3.26 + 3.26i)7-s + (−1.69 + 1.69i)8-s + (0.444 − 1.44i)10-s + (−1.89 + 2.72i)11-s + (−3.61 + 3.61i)13-s − 3.12i·14-s − 1.46·16-s + (0.0888 + 0.0888i)17-s − 4.80·19-s + (3.04 − 1.61i)20-s + (2.20 − 0.395i)22-s + (0.771 + 0.771i)23-s + ⋯
L(s)  = 1  + (−0.338 − 0.338i)2-s − 0.771i·4-s + (0.468 + 0.883i)5-s + (1.23 + 1.23i)7-s + (−0.599 + 0.599i)8-s + (0.140 − 0.457i)10-s + (−0.571 + 0.820i)11-s + (−1.00 + 1.00i)13-s − 0.834i·14-s − 0.365·16-s + (0.0215 + 0.0215i)17-s − 1.10·19-s + (0.681 − 0.361i)20-s + (0.470 − 0.0843i)22-s + (0.160 + 0.160i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.624 - 0.781i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.624 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07730 + 0.517996i\)
\(L(\frac12)\) \(\approx\) \(1.07730 + 0.517996i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.04 - 1.97i)T \)
11 \( 1 + (1.89 - 2.72i)T \)
good2 \( 1 + (0.478 + 0.478i)T + 2iT^{2} \)
7 \( 1 + (-3.26 - 3.26i)T + 7iT^{2} \)
13 \( 1 + (3.61 - 3.61i)T - 13iT^{2} \)
17 \( 1 + (-0.0888 - 0.0888i)T + 17iT^{2} \)
19 \( 1 + 4.80T + 19T^{2} \)
23 \( 1 + (-0.771 - 0.771i)T + 23iT^{2} \)
29 \( 1 - 4.30T + 29T^{2} \)
31 \( 1 - 2.71T + 31T^{2} \)
37 \( 1 + (-4.40 + 4.40i)T - 37iT^{2} \)
41 \( 1 + 6.12iT - 41T^{2} \)
43 \( 1 + (-1.72 + 1.72i)T - 43iT^{2} \)
47 \( 1 + (-7.28 + 7.28i)T - 47iT^{2} \)
53 \( 1 + (-0.661 - 0.661i)T + 53iT^{2} \)
59 \( 1 + 0.294iT - 59T^{2} \)
61 \( 1 + 4.60iT - 61T^{2} \)
67 \( 1 + (-1.64 + 1.64i)T - 67iT^{2} \)
71 \( 1 + 1.56T + 71T^{2} \)
73 \( 1 + (-5.15 + 5.15i)T - 73iT^{2} \)
79 \( 1 - 3.59T + 79T^{2} \)
83 \( 1 + (-1.26 + 1.26i)T - 83iT^{2} \)
89 \( 1 - 14.7iT - 89T^{2} \)
97 \( 1 + (-11.2 + 11.2i)T - 97iT^{2} \)
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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

−10.94927725892445944359614646015, −10.25314388585743744371381516117, −9.399755302518140105997163043266, −8.644071284942000993195987375236, −7.44429828245208915461918796045, −6.38958703103689214941150851470, −5.41148252213541409179080783378, −4.61691492173664047333591177786, −2.27397706707558123649562561352, −2.14709333061965145802745083580, 0.801209871916428094718445855349, 2.68690873381892838004860550352, 4.25467942749122853431578253126, 4.97490840454680955539624984166, 6.27578522147991005292776431350, 7.57306540309648749051000295634, 8.065872051254108410075190687613, 8.702827966713183594891620093524, 9.946965284044102412554263301549, 10.69776617488240333599822033617

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