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Properties

Label 2-221-13.10-c1-0-7
Degree $2$
Conductor $221$
Sign $0.897 + 0.441i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.482 + 0.278i)2-s + (−1.01 − 1.75i)3-s + (−0.844 + 1.46i)4-s + 0.0627i·5-s + (0.980 + 0.565i)6-s + (2.50 + 1.44i)7-s − 2.05i·8-s + (−0.562 + 0.974i)9-s + (−0.0174 − 0.0302i)10-s + (2.78 − 1.60i)11-s + 3.43·12-s + (1.55 − 3.25i)13-s − 1.61·14-s + (0.110 − 0.0637i)15-s + (−1.11 − 1.93i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.341 + 0.196i)2-s + (−0.586 − 1.01i)3-s + (−0.422 + 0.731i)4-s + 0.0280i·5-s + (0.400 + 0.230i)6-s + (0.945 + 0.546i)7-s − 0.726i·8-s + (−0.187 + 0.324i)9-s + (−0.00552 − 0.00957i)10-s + (0.839 − 0.484i)11-s + 0.990·12-s + (0.430 − 0.902i)13-s − 0.430·14-s + (0.0285 − 0.0164i)15-s + (−0.279 − 0.483i)16-s + (0.121 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $0.897 + 0.441i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 221,\ (\ :1/2),\ 0.897 + 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869387 - 0.202538i\)
\(L(\frac12)\) \(\approx\) \(0.869387 - 0.202538i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.55 + 3.25i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.482 - 0.278i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.01 + 1.75i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.0627iT - 5T^{2} \)
7 \( 1 + (-2.50 - 1.44i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.78 + 1.60i)T + (5.5 - 9.52i)T^{2} \)
19 \( 1 + (-5.12 - 2.96i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.06 + 5.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.42 - 5.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.67iT - 31T^{2} \)
37 \( 1 + (2.12 - 1.22i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.760 - 0.439i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.15 - 8.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 - 9.18T + 53T^{2} \)
59 \( 1 + (7.13 + 4.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.793 - 1.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.36 - 4.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.30 + 4.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.94iT - 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 5.01iT - 83T^{2} \)
89 \( 1 + (-1.96 + 1.13i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.84 - 2.21i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.08908102323005422726673300729, −11.68440621337615305201313313120, −10.32006023181835920052854543352, −8.859465166157298522405469545395, −8.227230758005016544551645194970, −7.25097725131898180978795177448, −6.23347222420307300018429273162, −4.99721869558924562425220448149, −3.31891382197373854014112908936, −1.16861593536514254712546928971, 1.47737374830975044657669539403, 4.13377584249484184641262687166, 4.79444382210835504264526709764, 5.87924505929485541199485751433, 7.36465851380146050142860507508, 8.792734416222039032153740744179, 9.628785719914533444539109290735, 10.34858977895445768250875352200, 11.30649859421351237194006167549, 11.73611078376851133589474511361

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