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Properties

Label 2-221-13.4-c1-0-9
Degree $2$
Conductor $221$
Sign $0.953 + 0.302i$
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  + (0.720 + 0.416i)2-s + (−0.251 + 0.434i)3-s + (−0.653 − 1.13i)4-s − 1.03i·5-s + (−0.361 + 0.208i)6-s + (2.58 − 1.49i)7-s − 2.75i·8-s + (1.37 + 2.37i)9-s + (0.430 − 0.746i)10-s + (1.81 + 1.04i)11-s + 0.656·12-s + (1.13 − 3.42i)13-s + 2.48·14-s + (0.450 + 0.260i)15-s + (−0.162 + 0.281i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.509 + 0.294i)2-s + (−0.144 + 0.251i)3-s + (−0.326 − 0.566i)4-s − 0.463i·5-s + (−0.147 + 0.0853i)6-s + (0.976 − 0.563i)7-s − 0.973i·8-s + (0.457 + 0.793i)9-s + (0.136 − 0.235i)10-s + (0.547 + 0.316i)11-s + 0.189·12-s + (0.314 − 0.949i)13-s + 0.663·14-s + (0.116 + 0.0671i)15-s + (−0.0405 + 0.0702i)16-s + (−0.121 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $0.953 + 0.302i$
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: \((2,\ 221,\ (\ :1/2),\ 0.953 + 0.302i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50718 - 0.233062i\)
\(L(\frac12)\) \(\approx\) \(1.50718 - 0.233062i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.13 + 3.42i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.720 - 0.416i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.251 - 0.434i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.03iT - 5T^{2} \)
7 \( 1 + (-2.58 + 1.49i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.81 - 1.04i)T + (5.5 + 9.52i)T^{2} \)
19 \( 1 + (4.93 - 2.84i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.255 - 0.442i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.13 - 7.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.19iT - 31T^{2} \)
37 \( 1 + (8.44 + 4.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.78 - 3.91i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.19 - 7.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.15iT - 47T^{2} \)
53 \( 1 - 0.563T + 53T^{2} \)
59 \( 1 + (5.15 - 2.97i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.86 + 3.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.6 - 7.31i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.86 - 3.38i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.54iT - 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 13.0iT - 83T^{2} \)
89 \( 1 + (-11.6 - 6.70i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.71 + 3.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.59225209916150756432230148577, −10.96484797699013315315159974461, −10.52559688979721546792182028094, −9.351261642208306422299545503341, −8.204036242785632805184632475394, −7.11208800860334026606757986534, −5.71075255056799795690529567809, −4.80990447719403613340269379895, −4.04975463716780303292928365613, −1.45254477215134361663280916322, 2.09238060733428372767521460564, 3.71491733951727069563853058130, 4.67686897333776229964176014552, 6.14502082495177956507799660166, 7.19324969507125550236036110120, 8.540010456439365797242638578091, 9.116274746236539648694552208406, 10.79604805919244224197736077243, 11.61073533266237576409557991858, 12.20620406393620797069512634538

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