↑

Properties

Label 2-495-11.9-c1-0-13
Degree $2$
Conductor $495$
Sign $0.322 + 0.946i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.456 − 1.40i)2-s + (−0.147 − 0.107i)4-s + (−0.309 − 0.951i)5-s + (1.85 + 1.34i)7-s + (2.17 − 1.57i)8-s − 1.47·10-s + (3.12 + 1.12i)11-s + (−0.661 + 2.03i)13-s + (2.74 − 1.99i)14-s + (−1.33 − 4.11i)16-s + (−0.168 − 0.517i)17-s + (1.76 − 1.28i)19-s + (−0.0563 + 0.173i)20-s + (3.00 − 3.87i)22-s − 2.03·23-s + ⋯
L(s)  = 1  + (0.322 − 0.993i)2-s + (−0.0737 − 0.0535i)4-s + (−0.138 − 0.425i)5-s + (0.701 + 0.509i)7-s + (0.768 − 0.558i)8-s − 0.467·10-s + (0.940 + 0.339i)11-s + (−0.183 + 0.564i)13-s + (0.733 − 0.532i)14-s + (−0.334 − 1.02i)16-s + (−0.0408 − 0.125i)17-s + (0.405 − 0.294i)19-s + (−0.0125 + 0.0387i)20-s + (0.640 − 0.825i)22-s − 0.425·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.322 + 0.946i$
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: \((2,\ 495,\ (\ :1/2),\ 0.322 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64914 - 1.18058i\)
\(L(\frac12)\) \(\approx\) \(1.64914 - 1.18058i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 - 1.12i)T \)
good2 \( 1 + (-0.456 + 1.40i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-1.85 - 1.34i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.661 - 2.03i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.168 + 0.517i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.76 + 1.28i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 2.03T + 23T^{2} \)
29 \( 1 + (8.04 + 5.84i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.09 + 6.44i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.13 - 5.18i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.47 - 1.07i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.620T + 43T^{2} \)
47 \( 1 + (0.305 - 0.222i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.58 - 11.0i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.53 + 4.74i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.69 - 8.29i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 9.75T + 67T^{2} \)
71 \( 1 + (-4.63 - 14.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.35 + 4.61i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.85 - 8.77i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.92 - 8.98i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.583T + 89T^{2} \)
97 \( 1 + (1.66 - 5.11i)T + (-78.4 - 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.26973659249949005894771393304, −9.860150341824155562004062718645, −9.294715229005002752165528243158, −8.095549154100924462618576810569, −7.22586124635685823574837815673, −5.97503421824814024918243428085, −4.63210273949497659793068046652, −3.98928810397549046898750913077, −2.50832379574616677203905654544, −1.44934387989206448595703062620, 1.63312135911564450983282560743, 3.47730043338747266846716572470, 4.64325045989184093689985335052, 5.64783113981168151417992520912, 6.54679979387657540436480810491, 7.43057921081469805409369114914, 8.027176713408964559028056199132, 9.170138556756500824383811523364, 10.44098861704624853404157126051, 11.04137706459972142221476393510

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