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Properties

Label 2-495-15.14-c2-0-17
Degree $2$
Conductor $495$
Sign $0.989 + 0.143i$
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  + 0.881·2-s − 3.22·4-s + (−4.45 + 2.27i)5-s − 4.31i·7-s − 6.36·8-s + (−3.92 + 2.00i)10-s + 3.31i·11-s − 0.805i·13-s − 3.80i·14-s + 7.27·16-s + 23.4·17-s + 24.5·19-s + (14.3 − 7.32i)20-s + 2.92i·22-s + 9.68·23-s + ⋯
L(s)  = 1  + 0.440·2-s − 0.805·4-s + (−0.890 + 0.454i)5-s − 0.617i·7-s − 0.795·8-s + (−0.392 + 0.200i)10-s + 0.301i·11-s − 0.0619i·13-s − 0.272i·14-s + 0.454·16-s + 1.37·17-s + 1.28·19-s + (0.717 − 0.366i)20-s + 0.132i·22-s + 0.421·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.989 + 0.143i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ 0.989 + 0.143i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.409902682\)
\(L(\frac12)\) \(\approx\) \(1.409902682\)
\(L(2)\) 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 \( 1 + (4.45 - 2.27i)T \)
11 \( 1 - 3.31iT \)
good2 \( 1 - 0.881T + 4T^{2} \)
7 \( 1 + 4.31iT - 49T^{2} \)
13 \( 1 + 0.805iT - 169T^{2} \)
17 \( 1 - 23.4T + 289T^{2} \)
19 \( 1 - 24.5T + 361T^{2} \)
23 \( 1 - 9.68T + 529T^{2} \)
29 \( 1 + 11.7iT - 841T^{2} \)
31 \( 1 + 26.1T + 961T^{2} \)
37 \( 1 + 16.0iT - 1.36e3T^{2} \)
41 \( 1 - 72.3iT - 1.68e3T^{2} \)
43 \( 1 + 44.7iT - 1.84e3T^{2} \)
47 \( 1 - 51.7T + 2.20e3T^{2} \)
53 \( 1 - 39.9T + 2.80e3T^{2} \)
59 \( 1 - 72.6iT - 3.48e3T^{2} \)
61 \( 1 + 51.9T + 3.72e3T^{2} \)
67 \( 1 + 80.0iT - 4.48e3T^{2} \)
71 \( 1 + 6.50iT - 5.04e3T^{2} \)
73 \( 1 + 48.8iT - 5.32e3T^{2} \)
79 \( 1 - 65.8T + 6.24e3T^{2} \)
83 \( 1 - 54.1T + 6.88e3T^{2} \)
89 \( 1 + 147. iT - 7.92e3T^{2} \)
97 \( 1 + 155. iT - 9.40e3T^{2} \)
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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

−10.69055592505958913427638175397, −9.886511713332852702613971723457, −8.960711382530115928385766785031, −7.77836823583007521573681024805, −7.30139975394540775082766388348, −5.87893604286603195107733698542, −4.86508204900991339725604341141, −3.85589523002169534840032736930, −3.12474686615086265660937045730, −0.78041274026352000394021776153, 0.889287789859818761146187371930, 3.11580566497311989732225271650, 3.92041301257513061001243232685, 5.17578639042284097535228492224, 5.63187236144744748103967702390, 7.24995844973887116556743073569, 8.144147175624898599116399490027, 8.965455042766787509636276446378, 9.638875790193559226534733821206, 10.89742325556579902897695995016

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