↑

Properties

Label 2-99-33.20-c2-0-1
Degree $2$
Conductor $99$
Sign $0.347 - 0.937i$
Analytic cond. $2.69755$
Root an. cond. $1.64242$
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  + (−1.28 − 0.417i)2-s + (−1.75 − 1.27i)4-s + (−4.85 + 1.57i)5-s + (10.6 + 7.73i)7-s + (4.90 + 6.74i)8-s + 6.89·10-s + (−4.16 + 10.1i)11-s + (−0.825 + 2.53i)13-s + (−10.4 − 14.3i)14-s + (−0.797 − 2.45i)16-s + (16.3 − 5.32i)17-s + (−24.7 + 17.9i)19-s + (10.5 + 3.42i)20-s + (9.60 − 11.3i)22-s + 15.5i·23-s + ⋯
L(s)  = 1  + (−0.642 − 0.208i)2-s + (−0.439 − 0.319i)4-s + (−0.970 + 0.315i)5-s + (1.52 + 1.10i)7-s + (0.613 + 0.843i)8-s + 0.689·10-s + (−0.378 + 0.925i)11-s + (−0.0634 + 0.195i)13-s + (−0.747 − 1.02i)14-s + (−0.0498 − 0.153i)16-s + (0.963 − 0.312i)17-s + (−1.30 + 0.947i)19-s + (0.527 + 0.171i)20-s + (0.436 − 0.515i)22-s + 0.675i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.347 - 0.937i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.347 - 0.937i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.570726 + 0.397083i\)
\(L(\frac12)\) \(\approx\) \(0.570726 + 0.397083i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (4.16 - 10.1i)T \)
good2 \( 1 + (1.28 + 0.417i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (4.85 - 1.57i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (-10.6 - 7.73i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (0.825 - 2.53i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-16.3 + 5.32i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (24.7 - 17.9i)T + (111. - 343. i)T^{2} \)
23 \( 1 - 15.5iT - 529T^{2} \)
29 \( 1 + (-6.10 + 8.39i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-5.21 + 16.0i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (15.1 + 11.0i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (7.85 + 10.8i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 10.5T + 1.84e3T^{2} \)
47 \( 1 + (-48.3 - 66.5i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (59.9 + 19.4i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-38.7 + 53.3i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-11.2 - 34.7i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 60.5T + 4.48e3T^{2} \)
71 \( 1 + (-46.7 + 15.2i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (5.15 + 3.74i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-35.5 + 109. i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (38.5 - 12.5i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 71.1iT - 7.92e3T^{2} \)
97 \( 1 + (-5.76 + 17.7i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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

−14.22584382391783977476661619967, −12.45988791751742686976778018546, −11.60014726872879623687770099792, −10.70080370248364341462973143191, −9.473316217747080047383069207042, −8.232068082585866043272303368150, −7.71304089494360157426984110530, −5.51860395411650914308709761684, −4.38155643112180013974818082943, −1.94408966670509740213623702320, 0.70916876387555436827189614100, 3.87773168901917486147091708381, 4.86580324367646967836568583429, 7.14633851849423849525558739916, 8.214569578186437624662550365158, 8.467666260829541460268249971051, 10.34928795047967925128616794566, 11.11374383491921189504892241359, 12.35932771928343899755194465438, 13.52285114833620801320663394788

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