↑

Properties

Label 2-135-5.3-c2-0-0
Degree $2$
Conductor $135$
Sign $-0.796 + 0.604i$
Analytic cond. $3.67848$
Root an. cond. $1.91793$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.05 + 1.05i)2-s + 1.78i·4-s + (−2.02 − 4.57i)5-s + (−5.20 + 5.20i)7-s + (−6.08 − 6.08i)8-s + (6.94 + 2.67i)10-s − 6.54·11-s + (−5.46 − 5.46i)13-s − 10.9i·14-s + 5.65·16-s + (−10.2 + 10.2i)17-s − 23.3i·19-s + (8.16 − 3.62i)20-s + (6.88 − 6.88i)22-s + (−25.1 − 25.1i)23-s + ⋯
L(s)  = 1  + (−0.525 + 0.525i)2-s + 0.446i·4-s + (−0.405 − 0.914i)5-s + (−0.742 + 0.742i)7-s + (−0.760 − 0.760i)8-s + (0.694 + 0.267i)10-s − 0.595·11-s + (−0.420 − 0.420i)13-s − 0.781i·14-s + 0.353·16-s + (−0.605 + 0.605i)17-s − 1.22i·19-s + (0.408 − 0.181i)20-s + (0.313 − 0.313i)22-s + (−1.09 − 1.09i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.796 + 0.604i$
Analytic conductor: \(3.67848\)
Root analytic conductor: \(1.91793\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1),\ -0.796 + 0.604i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0124928 - 0.0371113i\)
\(L(\frac12)\) \(\approx\) \(0.0124928 - 0.0371113i\)
\(L(2)\) 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 \( 1 + (2.02 + 4.57i)T \)
good2 \( 1 + (1.05 - 1.05i)T - 4iT^{2} \)
7 \( 1 + (5.20 - 5.20i)T - 49iT^{2} \)
11 \( 1 + 6.54T + 121T^{2} \)
13 \( 1 + (5.46 + 5.46i)T + 169iT^{2} \)
17 \( 1 + (10.2 - 10.2i)T - 289iT^{2} \)
19 \( 1 + 23.3iT - 361T^{2} \)
23 \( 1 + (25.1 + 25.1i)T + 529iT^{2} \)
29 \( 1 - 55.5iT - 841T^{2} \)
31 \( 1 - 25.5T + 961T^{2} \)
37 \( 1 + (28.4 - 28.4i)T - 1.36e3iT^{2} \)
41 \( 1 - 22.0T + 1.68e3T^{2} \)
43 \( 1 + (-5.36 - 5.36i)T + 1.84e3iT^{2} \)
47 \( 1 + (25.6 - 25.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (-33.7 - 33.7i)T + 2.80e3iT^{2} \)
59 \( 1 - 26.5iT - 3.48e3T^{2} \)
61 \( 1 - 66.0T + 3.72e3T^{2} \)
67 \( 1 + (28.3 - 28.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 81.4T + 5.04e3T^{2} \)
73 \( 1 + (91.4 + 91.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 151. iT - 6.24e3T^{2} \)
83 \( 1 + (62.8 + 62.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 92.2iT - 7.92e3T^{2} \)
97 \( 1 + (-21.6 + 21.6i)T - 9.40e3iT^{2} \)
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

−13.20453743308166020381528239996, −12.63873930334410392614281696938, −11.87469849189711266876353338843, −10.28556554956593796345567710239, −8.988590810313336282776682026343, −8.532203065199502051354776965384, −7.34949196596679150161761157213, −6.11779456429240256675045463619, −4.62656188998722665379092825704, −2.93522638273575657686047789502, 0.02893694445200124673160381496, 2.37481449501012615445336641315, 3.91382329886745771577181772815, 5.82140159921362656094575868366, 6.98571064329408451381356821356, 8.127519231490227154020928271208, 9.842429132257022429135705384627, 10.05784894865678135811281540331, 11.23810367226833325413976355565, 11.99778247199359362274838471108

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