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Properties

Label 2-495-45.23-c1-0-32
Degree $2$
Conductor $495$
Sign $0.993 - 0.110i$
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  + (−0.00736 + 0.0274i)2-s + (0.724 − 1.57i)3-s + (1.73 + 0.999i)4-s + (0.674 + 2.13i)5-s + (0.0378 + 0.0314i)6-s + (3.09 + 0.830i)7-s + (−0.0804 + 0.0804i)8-s + (−1.94 − 2.28i)9-s + (−0.0635 + 0.00284i)10-s + (0.866 − 0.5i)11-s + (2.82 − 1.99i)12-s + (−2.89 + 0.776i)13-s + (−0.0456 + 0.0790i)14-s + (3.84 + 0.483i)15-s + (1.99 + 3.45i)16-s + (−3.59 − 3.59i)17-s + ⋯
L(s)  = 1  + (−0.00520 + 0.0194i)2-s + (0.418 − 0.908i)3-s + (0.865 + 0.499i)4-s + (0.301 + 0.953i)5-s + (0.0154 + 0.0128i)6-s + (1.17 + 0.313i)7-s + (−0.0284 + 0.0284i)8-s + (−0.649 − 0.760i)9-s + (−0.0200 + 0.000900i)10-s + (0.261 − 0.150i)11-s + (0.816 − 0.577i)12-s + (−0.804 + 0.215i)13-s + (−0.0121 + 0.0211i)14-s + (0.992 + 0.124i)15-s + (0.499 + 0.864i)16-s + (−0.872 − 0.872i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.993 - 0.110i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15197 + 0.119792i\)
\(L(\frac12)\) \(\approx\) \(2.15197 + 0.119792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.724 + 1.57i)T \)
5 \( 1 + (-0.674 - 2.13i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (0.00736 - 0.0274i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (-3.09 - 0.830i)T + (6.06 + 3.5i)T^{2} \)
13 \( 1 + (2.89 - 0.776i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (3.59 + 3.59i)T + 17iT^{2} \)
19 \( 1 - 7.26iT - 19T^{2} \)
23 \( 1 + (0.0589 + 0.219i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.30 + 3.99i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.22 + 5.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.56 + 6.56i)T - 37iT^{2} \)
41 \( 1 + (1.51 + 0.874i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.67 + 6.25i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.00 + 11.2i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.30 - 4.30i)T - 53iT^{2} \)
59 \( 1 + (1.55 - 2.69i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.46 + 6.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.867 + 3.23i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.40iT - 71T^{2} \)
73 \( 1 + (-1.17 - 1.17i)T + 73iT^{2} \)
79 \( 1 + (6.40 - 3.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.77 + 2.08i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 7.03T + 89T^{2} \)
97 \( 1 + (-8.71 - 2.33i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.39472756377144144574179654052, −10.18001618179984965244791910828, −8.993188214739623597760732658633, −7.86219076526983760533937675327, −7.49185810502990337077268022144, −6.52773885796716205378463266645, −5.69123591947693880262570830957, −3.88606175469658782747803693964, −2.49742665385471139529568124029, −1.96604687906889006149130158334, 1.53019181397467405084183222995, 2.74680559544290192593253242101, 4.63320907882974084081719229938, 4.84396722143549343654666050645, 6.15120665645739262328578326662, 7.44310538093651247024191928902, 8.377500723686159754995103200243, 9.215810674332307575290404679125, 10.04615919376232470687527580494, 10.98345191233997359644545508596

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