↑

Properties

Label 2-90-15.2-c11-0-19
Degree $2$
Conductor $90$
Sign $-0.896 - 0.442i$
Analytic cond. $69.1508$
Root an. cond. $8.31570$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−22.6 + 22.6i)2-s − 1.02e3i·4-s + (−6.46e3 + 2.65e3i)5-s + (−5.94e4 − 5.94e4i)7-s + (2.31e4 + 2.31e4i)8-s + (8.62e4 − 2.06e5i)10-s − 1.00e6i·11-s + (−3.29e5 + 3.29e5i)13-s + 2.68e6·14-s − 1.04e6·16-s + (6.44e6 − 6.44e6i)17-s + 1.74e5i·19-s + (2.71e6 + 6.62e6i)20-s + (2.28e7 + 2.28e7i)22-s + (−2.76e7 − 2.76e7i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.925 + 0.379i)5-s + (−1.33 − 1.33i)7-s + (0.250 + 0.250i)8-s + (0.272 − 0.652i)10-s − 1.88i·11-s + (−0.246 + 0.246i)13-s + 1.33·14-s − 0.250·16-s + (1.10 − 1.10i)17-s + 0.0161i·19-s + (0.189 + 0.462i)20-s + (0.943 + 0.943i)22-s + (−0.896 − 0.896i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.896 - 0.442i$
Analytic conductor: \(69.1508\)
Root analytic conductor: \(8.31570\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :11/2),\ -0.896 - 0.442i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.104886 + 0.449973i\)
\(L(\frac12)\) \(\approx\) \(0.104886 + 0.449973i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (22.6 - 22.6i)T \)
3 \( 1 \)
5 \( 1 + (6.46e3 - 2.65e3i)T \)
good7 \( 1 + (5.94e4 + 5.94e4i)T + 1.97e9iT^{2} \)
11 \( 1 + 1.00e6iT - 2.85e11T^{2} \)
13 \( 1 + (3.29e5 - 3.29e5i)T - 1.79e12iT^{2} \)
17 \( 1 + (-6.44e6 + 6.44e6i)T - 3.42e13iT^{2} \)
19 \( 1 - 1.74e5iT - 1.16e14T^{2} \)
23 \( 1 + (2.76e7 + 2.76e7i)T + 9.52e14iT^{2} \)
29 \( 1 + 9.08e7T + 1.22e16T^{2} \)
31 \( 1 + 1.71e7T + 2.54e16T^{2} \)
37 \( 1 + (2.62e8 + 2.62e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 6.49e8iT - 5.50e17T^{2} \)
43 \( 1 + (-8.54e8 + 8.54e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (-6.21e8 + 6.21e8i)T - 2.47e18iT^{2} \)
53 \( 1 + (-5.04e8 - 5.04e8i)T + 9.26e18iT^{2} \)
59 \( 1 + 1.95e9T + 3.01e19T^{2} \)
61 \( 1 - 2.35e9T + 4.35e19T^{2} \)
67 \( 1 + (9.94e9 + 9.94e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 2.25e10iT - 2.31e20T^{2} \)
73 \( 1 + (2.94e9 - 2.94e9i)T - 3.13e20iT^{2} \)
79 \( 1 - 6.53e9iT - 7.47e20T^{2} \)
83 \( 1 + (3.76e10 + 3.76e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 8.81e10T + 2.77e21T^{2} \)
97 \( 1 + (3.37e10 + 3.37e10i)T + 7.15e21iT^{2} \)
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

−10.98222895067844959178108796998, −10.22621868936433835683394171039, −8.989676819181071838141063585971, −7.71041251478365199677087390811, −6.96477930990458436375405335036, −5.83939910286343567093634451101, −3.95303882398901421005941577614, −3.09834614517889000418879675916, −0.58691043885881229202790211627, −0.24510588871878396236898317229, 1.59975763648405736618698796401, 2.91869512831405210265199477577, 4.08250648085182594513968183486, 5.65243711999508647472452903450, 7.17290680537403700972497272134, 8.221334824751301625905135338395, 9.455887354974351627161108996909, 10.02316051975194150440251421714, 11.69190920654449120794916845473, 12.51781548135995779268005686549

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