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Properties

Label 2-495-11.5-c1-0-15
Degree $2$
Conductor $495$
Sign $-0.102 + 0.994i$
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  + (−0.147 − 0.453i)2-s + (1.43 − 1.04i)4-s + (0.309 − 0.951i)5-s + (−2.17 + 1.57i)7-s + (−1.45 − 1.05i)8-s − 0.477·10-s + (2.79 − 1.79i)11-s + (−1.44 − 4.43i)13-s + (1.03 + 0.753i)14-s + (0.829 − 2.55i)16-s + (1.42 − 4.39i)17-s + (3.51 + 2.55i)19-s + (−0.547 − 1.68i)20-s + (−1.22 − 1.00i)22-s − 2.77·23-s + ⋯
L(s)  = 1  + (−0.104 − 0.320i)2-s + (0.716 − 0.520i)4-s + (0.138 − 0.425i)5-s + (−0.821 + 0.596i)7-s + (−0.514 − 0.374i)8-s − 0.150·10-s + (0.841 − 0.540i)11-s + (−0.400 − 1.23i)13-s + (0.277 + 0.201i)14-s + (0.207 − 0.638i)16-s + (0.346 − 1.06i)17-s + (0.805 + 0.585i)19-s + (−0.122 − 0.376i)20-s + (−0.261 − 0.213i)22-s − 0.578·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.102 + 0.994i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.102 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967630 - 1.07285i\)
\(L(\frac12)\) \(\approx\) \(0.967630 - 1.07285i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.79 + 1.79i)T \)
good2 \( 1 + (0.147 + 0.453i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (2.17 - 1.57i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.44 + 4.43i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.42 + 4.39i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.51 - 2.55i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.77T + 23T^{2} \)
29 \( 1 + (2.43 - 1.77i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.737 - 2.26i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-8.61 + 6.25i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.78 + 1.29i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.06T + 43T^{2} \)
47 \( 1 + (-3.52 - 2.56i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.95 - 6.02i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.50 + 6.90i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.23 - 3.78i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 7.31T + 67T^{2} \)
71 \( 1 + (0.369 - 1.13i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.826 - 0.600i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.08 - 3.33i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.43 - 10.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 2.76T + 89T^{2} \)
97 \( 1 + (-5.72 - 17.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.66763268388495598790858530613, −9.687471332733307863851453474912, −9.336467705360883341336075166057, −8.022233006197871965529636776672, −6.94656830127388112494252257891, −5.90162951836244786399026923520, −5.35702546637719896788161027378, −3.51504936140928946010557258950, −2.56467161078997870323584653659, −0.916299947081350534157515228896, 1.95033929819423362466787554991, 3.32031846519806829672898571602, 4.25650029190802915070022602408, 6.00122892204330864204740293548, 6.76232281263462339677345820461, 7.25773282904304742568650469108, 8.372382631216391438943315221553, 9.560311646615585573433048308342, 10.10899537603713586299216456171, 11.42186589039105603280022689804

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