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Properties

Label 2-221-13.4-c1-0-5
Degree $2$
Conductor $221$
Sign $0.989 + 0.141i$
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  + (−0.990 − 0.572i)2-s + (0.440 − 0.763i)3-s + (−0.345 − 0.598i)4-s + 4.00i·5-s + (−0.873 + 0.504i)6-s + (2.20 − 1.27i)7-s + 3.07i·8-s + (1.11 + 1.92i)9-s + (2.28 − 3.96i)10-s + (2.00 + 1.15i)11-s − 0.609·12-s + (0.556 − 3.56i)13-s − 2.91·14-s + (3.05 + 1.76i)15-s + (1.07 − 1.85i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.700 − 0.404i)2-s + (0.254 − 0.440i)3-s + (−0.172 − 0.299i)4-s + 1.79i·5-s + (−0.356 + 0.205i)6-s + (0.833 − 0.481i)7-s + 1.08i·8-s + (0.370 + 0.641i)9-s + (0.724 − 1.25i)10-s + (0.604 + 0.349i)11-s − 0.175·12-s + (0.154 − 0.988i)13-s − 0.778·14-s + (0.789 + 0.455i)15-s + (0.267 − 0.463i)16-s + (−0.121 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $0.989 + 0.141i$
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: \((2,\ 221,\ (\ :1/2),\ 0.989 + 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.972450 - 0.0692143i\)
\(L(\frac12)\) \(\approx\) \(0.972450 - 0.0692143i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.556 + 3.56i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.990 + 0.572i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.440 + 0.763i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 4.00iT - 5T^{2} \)
7 \( 1 + (-2.20 + 1.27i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.00 - 1.15i)T + (5.5 + 9.52i)T^{2} \)
19 \( 1 + (-1.88 + 1.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.652 - 1.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.93 + 3.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.83iT - 31T^{2} \)
37 \( 1 + (-6.21 - 3.58i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.43 + 2.56i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.95 - 3.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.28iT - 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + (0.723 - 0.417i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.54 + 7.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.5 + 6.64i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.916 - 0.529i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.95iT - 73T^{2} \)
79 \( 1 - 7.80T + 79T^{2} \)
83 \( 1 + 2.32iT - 83T^{2} \)
89 \( 1 + (-1.79 - 1.03i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.35 + 3.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.87004153887507056044569319424, −10.88466061370838550745274333086, −10.53351264110984036506314417336, −9.596322635630342800792487510717, −8.093016478718893807536453613982, −7.49260059389312416007772495095, −6.37886937946341327100722434327, −4.82298070684918344922367473929, −2.99405074123929645311766137483, −1.66744496023607139272122436645, 1.26367336396354373132790590537, 3.96539222455499480090683310845, 4.67610797562295516563809515542, 6.17869705341788279148135304739, 7.71895842170946559710697742229, 8.657114570432324477145839557484, 9.071599089795789459477846855426, 9.753693884035813502828383058941, 11.55053985557251745357258200531, 12.27548582770311110904293525449

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