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Properties

Label 2-495-11.9-c1-0-14
Degree $2$
Conductor $495$
Sign $-0.295 + 0.955i$
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  + (0.647 − 1.99i)2-s + (−1.93 − 1.40i)4-s + (0.309 + 0.951i)5-s + (2.48 + 1.80i)7-s + (−0.661 + 0.480i)8-s + 2.09·10-s + (−1.86 − 2.74i)11-s + (0.942 − 2.90i)13-s + (5.19 − 3.77i)14-s + (−0.947 − 2.91i)16-s + (0.143 + 0.441i)17-s + (6.38 − 4.64i)19-s + (0.738 − 2.27i)20-s + (−6.67 + 1.93i)22-s + 1.39·23-s + ⋯
L(s)  = 1  + (0.457 − 1.40i)2-s + (−0.966 − 0.702i)4-s + (0.138 + 0.425i)5-s + (0.937 + 0.681i)7-s + (−0.233 + 0.169i)8-s + 0.662·10-s + (−0.561 − 0.827i)11-s + (0.261 − 0.804i)13-s + (1.38 − 1.00i)14-s + (−0.236 − 0.729i)16-s + (0.0347 + 0.106i)17-s + (1.46 − 1.06i)19-s + (0.165 − 0.508i)20-s + (−1.42 + 0.412i)22-s + 0.289·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.295 + 0.955i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.295 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18907 - 1.61270i\)
\(L(\frac12)\) \(\approx\) \(1.18907 - 1.61270i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (1.86 + 2.74i)T \)
good2 \( 1 + (-0.647 + 1.99i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-2.48 - 1.80i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.942 + 2.90i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.143 - 0.441i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-6.38 + 4.64i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.39T + 23T^{2} \)
29 \( 1 + (-3.01 - 2.18i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.23 - 9.96i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.49 + 1.08i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.56 - 2.59i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.31T + 43T^{2} \)
47 \( 1 + (2.41 - 1.75i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.29 - 3.98i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.27 - 1.65i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.623 + 1.91i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 6.75T + 67T^{2} \)
71 \( 1 + (-2.01 - 6.20i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.98 - 5.80i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.57 - 10.9i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.75 + 8.48i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 + (-4.74 + 14.6i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.91726195839868061275078064749, −10.21050950481383403691744613589, −9.094324084532405694144418587983, −8.189979674216603177371811557585, −7.04493489049009243409116579896, −5.46900503240007531008533910202, −4.95173576266785146609990376796, −3.33313473975493291790374186883, −2.73065745377866562533133380318, −1.28033697268352633502463631169, 1.73466695167471545307450315076, 3.95395559248255867779705754410, 4.81745683439245230124895410401, 5.52724011700784726491930920529, 6.66778098590643608971484505115, 7.65698171188230473084605891022, 7.976722281291931384877839247756, 9.217306780805132069694784246224, 10.19525546525532974971036896770, 11.29995574370378126235695816677

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