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Properties

Label 2-495-11.9-c1-0-17
Degree $2$
Conductor $495$
Sign $-0.925 + 0.378i$
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  + (0.633 − 1.94i)2-s + (−1.77 − 1.29i)4-s + (−0.309 − 0.951i)5-s + (0.477 + 0.346i)7-s + (−0.329 + 0.239i)8-s − 2.04·10-s + (0.933 − 3.18i)11-s + (0.465 − 1.43i)13-s + (0.977 − 0.710i)14-s + (−1.10 − 3.38i)16-s + (−1.52 − 4.67i)17-s + (−5.51 + 4.00i)19-s + (−0.679 + 2.09i)20-s + (−5.61 − 3.83i)22-s + 0.0822·23-s + ⋯
L(s)  = 1  + (0.447 − 1.37i)2-s + (−0.889 − 0.646i)4-s + (−0.138 − 0.425i)5-s + (0.180 + 0.131i)7-s + (−0.116 + 0.0845i)8-s − 0.647·10-s + (0.281 − 0.959i)11-s + (0.128 − 0.396i)13-s + (0.261 − 0.189i)14-s + (−0.275 − 0.847i)16-s + (−0.368 − 1.13i)17-s + (−1.26 + 0.919i)19-s + (−0.151 + 0.467i)20-s + (−1.19 − 0.817i)22-s + 0.0171·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.925 + 0.378i$
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: \((2,\ 495,\ (\ :1/2),\ -0.925 + 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.328426 - 1.66859i\)
\(L(\frac12)\) \(\approx\) \(0.328426 - 1.66859i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.933 + 3.18i)T \)
good2 \( 1 + (-0.633 + 1.94i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-0.477 - 0.346i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.465 + 1.43i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.52 + 4.67i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.51 - 4.00i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 0.0822T + 23T^{2} \)
29 \( 1 + (-6.81 - 4.95i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.611 - 1.88i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.89 - 2.10i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.29 + 2.39i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.86T + 43T^{2} \)
47 \( 1 + (-4.67 + 3.39i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.09 - 6.44i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.51 + 3.28i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.296 + 0.913i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + (-0.396 - 1.22i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-12.6 - 9.18i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.44 - 10.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.23 - 3.81i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 6.08T + 89T^{2} \)
97 \( 1 + (2.07 - 6.37i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.85546914923144884020228489005, −9.943807353524051021373631205781, −8.934642110149064936939344319245, −8.148397720515950481611928156088, −6.77978355753936166019833037612, −5.50252505697123512704306888932, −4.49927159198219819184988128332, −3.52300933228338007894234872978, −2.41633769180542369358220916466, −0.940383772312052180825375018505, 2.18666601124313512713158963370, 4.13873611625671891162008408824, 4.64206089197555241600274232989, 6.13019638813859426984772369586, 6.58689182485778014604540544010, 7.53210841529763688948554999409, 8.315001524796167503040082435580, 9.298348024563811246108465434674, 10.53021594469875549575081179757, 11.22847340391220524344322210186

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