↑

Properties

Label 2-55-5.2-c2-0-6
Degree $2$
Conductor $55$
Sign $0.716 + 0.697i$
Analytic cond. $1.49864$
Root an. cond. $1.22419$
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  + (0.188 + 0.188i)2-s + (0.812 − 0.812i)3-s − 3.92i·4-s + (−1.21 − 4.84i)5-s + 0.305·6-s + (5.94 + 5.94i)7-s + (1.49 − 1.49i)8-s + 7.67i·9-s + (0.683 − 1.14i)10-s − 3.31·11-s + (−3.19 − 3.19i)12-s + (3.02 − 3.02i)13-s + 2.23i·14-s + (−4.92 − 2.95i)15-s − 15.1·16-s + (0.878 + 0.878i)17-s + ⋯
L(s)  = 1  + (0.0940 + 0.0940i)2-s + (0.270 − 0.270i)3-s − 0.982i·4-s + (−0.243 − 0.969i)5-s + 0.0509·6-s + (0.849 + 0.849i)7-s + (0.186 − 0.186i)8-s + 0.853i·9-s + (0.0683 − 0.114i)10-s − 0.301·11-s + (−0.266 − 0.266i)12-s + (0.232 − 0.232i)13-s + 0.159i·14-s + (−0.328 − 0.196i)15-s − 0.947·16-s + (0.0516 + 0.0516i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(1.49864\)
Root analytic conductor: \(1.22419\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1),\ 0.716 + 0.697i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.21584 - 0.493797i\)
\(L(\frac12)\) \(\approx\) \(1.21584 - 0.493797i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.21 + 4.84i)T \)
11 \( 1 + 3.31T \)
good2 \( 1 + (-0.188 - 0.188i)T + 4iT^{2} \)
3 \( 1 + (-0.812 + 0.812i)T - 9iT^{2} \)
7 \( 1 + (-5.94 - 5.94i)T + 49iT^{2} \)
13 \( 1 + (-3.02 + 3.02i)T - 169iT^{2} \)
17 \( 1 + (-0.878 - 0.878i)T + 289iT^{2} \)
19 \( 1 - 29.2iT - 361T^{2} \)
23 \( 1 + (-11.8 + 11.8i)T - 529iT^{2} \)
29 \( 1 + 15.7iT - 841T^{2} \)
31 \( 1 + 14.3T + 961T^{2} \)
37 \( 1 + (-36.1 - 36.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 57.6T + 1.68e3T^{2} \)
43 \( 1 + (-58.0 + 58.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (29.8 + 29.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (59.0 - 59.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 61.7iT - 3.48e3T^{2} \)
61 \( 1 - 1.59T + 3.72e3T^{2} \)
67 \( 1 + (32.7 + 32.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 12.4T + 5.04e3T^{2} \)
73 \( 1 + (74.3 - 74.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 21.8iT - 6.24e3T^{2} \)
83 \( 1 + (-45.9 + 45.9i)T - 6.88e3iT^{2} \)
89 \( 1 + 89.4iT - 7.92e3T^{2} \)
97 \( 1 + (62.6 + 62.6i)T + 9.40e3iT^{2} \)
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

−14.91236514104645012886838623867, −13.89562752488202578059572719019, −12.77653972500990229084946411204, −11.53225653797741929448524672565, −10.26162891416412529553872673071, −8.776852003322868304135469040473, −7.86275404361550842304064051547, −5.75530122640673775529865152377, −4.79793864925451289393078551130, −1.76683815116661996695888395739, 3.08384559203806056997747944258, 4.36525013173734177040317102577, 6.85931778670459618456055619339, 7.80201233235856823630566270415, 9.214108985083621446152743963159, 10.88201903857509299940240571856, 11.55016993275592071745706803912, 13.04694667949253652209071881631, 14.15119820644746608436382645840, 15.08937587346757787245243660242

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