↑

Properties

Label 2-135-9.5-c2-0-6
Degree $2$
Conductor $135$
Sign $0.358 + 0.933i$
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  + (2.83 − 1.63i)2-s + (3.34 − 5.79i)4-s + (1.93 + 1.11i)5-s + (−3.16 − 5.48i)7-s − 8.82i·8-s + 7.31·10-s + (12.8 − 7.43i)11-s + (−7.73 + 13.4i)13-s + (−17.9 − 10.3i)14-s + (−1.03 − 1.78i)16-s + 20.0i·17-s − 25.1·19-s + (12.9 − 7.48i)20-s + (24.3 − 42.1i)22-s + (−1.40 − 0.812i)23-s + ⋯
L(s)  = 1  + (1.41 − 0.817i)2-s + (0.837 − 1.44i)4-s + (0.387 + 0.223i)5-s + (−0.452 − 0.784i)7-s − 1.10i·8-s + 0.731·10-s + (1.17 − 0.676i)11-s + (−0.595 + 1.03i)13-s + (−1.28 − 0.740i)14-s + (−0.0644 − 0.111i)16-s + 1.18i·17-s − 1.32·19-s + (0.648 − 0.374i)20-s + (1.10 − 1.91i)22-s + (−0.0612 − 0.0353i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(3.67848\)
Root analytic conductor: \(1.91793\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1),\ 0.358 + 0.933i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.41280 - 1.65893i\)
\(L(\frac12)\) \(\approx\) \(2.41280 - 1.65893i\)
\(L(2)\) 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 \( 1 + (-1.93 - 1.11i)T \)
good2 \( 1 + (-2.83 + 1.63i)T + (2 - 3.46i)T^{2} \)
7 \( 1 + (3.16 + 5.48i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-12.8 + 7.43i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (7.73 - 13.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 20.0iT - 289T^{2} \)
19 \( 1 + 25.1T + 361T^{2} \)
23 \( 1 + (1.40 + 0.812i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-1.07 + 0.619i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (6.69 - 11.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 3.89T + 1.36e3T^{2} \)
41 \( 1 + (50.3 + 29.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-13.6 - 23.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-54.2 + 31.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 18.2iT - 2.80e3T^{2} \)
59 \( 1 + (-25.1 - 14.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (55.5 + 96.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-8.56 + 14.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 52.7iT - 5.04e3T^{2} \)
73 \( 1 + 71.0T + 5.32e3T^{2} \)
79 \( 1 + (30.2 + 52.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (70.2 - 40.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 6.34iT - 7.92e3T^{2} \)
97 \( 1 + (-7.84 - 13.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−12.83981658098143520880434963154, −11.97406564212803415257815485629, −10.95319005062621008859337090000, −10.17988621481251375794182535765, −8.802698678258642388090747615224, −6.79695617178753552107121652646, −6.03237862159473892869530109225, −4.39629637406682174232271841692, −3.56825406568326539368783622746, −1.86305429507948945339611640301, 2.70760859498661409125149846558, 4.29045889275652720496746422104, 5.41300607211778765630956143019, 6.35967689049178337492360479462, 7.36134958072582534278377504073, 8.895013152667177640320221193663, 9.993860195123726687462922743057, 11.77474055207315195515133659396, 12.48903019749548170057763039901, 13.19402165006225056006004998572

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