↑

Properties

Label 2-221-13.4-c1-0-17
Degree $2$
Conductor $221$
Sign $-0.695 + 0.718i$
Analytic cond. $1.76469$
Root an. cond. $1.32841$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.544 − 0.314i)2-s + (1.63 − 2.83i)3-s + (−0.802 − 1.38i)4-s − 0.315i·5-s + (−1.78 + 1.03i)6-s + (−0.424 + 0.244i)7-s + 2.26i·8-s + (−3.87 − 6.70i)9-s + (−0.0991 + 0.171i)10-s + (3.92 + 2.26i)11-s − 5.25·12-s + (2.55 + 2.54i)13-s + 0.308·14-s + (−0.895 − 0.516i)15-s + (−0.892 + 1.54i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.385 − 0.222i)2-s + (0.946 − 1.63i)3-s + (−0.401 − 0.694i)4-s − 0.141i·5-s + (−0.728 + 0.420i)6-s + (−0.160 + 0.0925i)7-s + 0.801i·8-s + (−1.29 − 2.23i)9-s + (−0.0313 + 0.0543i)10-s + (1.18 + 0.682i)11-s − 1.51·12-s + (0.709 + 0.704i)13-s + 0.0823·14-s + (−0.231 − 0.133i)15-s + (−0.223 + 0.386i)16-s + (−0.121 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $-0.695 + 0.718i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 221,\ (\ :1/2),\ -0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.471394 - 1.11306i\)
\(L(\frac12)\) \(\approx\) \(0.471394 - 1.11306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.55 - 2.54i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.544 + 0.314i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.63 + 2.83i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.315iT - 5T^{2} \)
7 \( 1 + (0.424 - 0.244i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.92 - 2.26i)T + (5.5 + 9.52i)T^{2} \)
19 \( 1 + (6.11 - 3.53i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.82 + 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.00 + 1.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.65iT - 31T^{2} \)
37 \( 1 + (-2.52 - 1.45i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.36 + 3.09i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.23 - 3.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.456iT - 47T^{2} \)
53 \( 1 + 4.92T + 53T^{2} \)
59 \( 1 + (-0.127 + 0.0738i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.76 - 11.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.106 + 0.0611i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-10.2 + 5.89i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.45iT - 73T^{2} \)
79 \( 1 - 0.666T + 79T^{2} \)
83 \( 1 - 5.05iT - 83T^{2} \)
89 \( 1 + (-8.65 - 4.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.50 - 5.49i)T + (48.5 - 84.0i)T^{2} \)
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

−12.13483180286354402349747089327, −11.06789824461748571914182555224, −9.576602342355176307401710808475, −8.854239766135424418934841194547, −8.201828292158389458356664886675, −6.74802171344900327033421934925, −6.22604687807552024785535830105, −4.19187906323653294707993206629, −2.29700595082472380911235865310, −1.21490991255203133836497477634, 3.16802241155600352079212287893, 3.78688682653134856284831483078, 4.94469013720544920105004346791, 6.69335179362846966938176687200, 8.261130917461674488492259611079, 8.763006669948113836111207265303, 9.416987274185372405948626169440, 10.52584503041615587591005627361, 11.26489075659676330270119796825, 12.87681843185109610641412191683

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