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Properties

Label 2-221-221.5-c1-0-10
Degree $2$
Conductor $221$
Sign $0.927 - 0.374i$
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  + (2.09 + 0.869i)2-s + (−0.864 − 1.29i)3-s + (2.23 + 2.23i)4-s + (0.867 + 1.29i)5-s + (−0.690 − 3.46i)6-s + (0.864 − 1.29i)7-s + (1.01 + 2.44i)8-s + (0.220 − 0.532i)9-s + (0.692 + 3.47i)10-s + (1.12 + 5.63i)11-s + (0.961 − 4.83i)12-s + (−2.75 − 2.32i)13-s + (2.94 − 1.96i)14-s + (0.929 − 2.24i)15-s − 0.313i·16-s + (−3.48 + 2.19i)17-s + ⋯
L(s)  = 1  + (1.48 + 0.614i)2-s + (−0.499 − 0.747i)3-s + (1.11 + 1.11i)4-s + (0.387 + 0.580i)5-s + (−0.281 − 1.41i)6-s + (0.326 − 0.489i)7-s + (0.358 + 0.864i)8-s + (0.0735 − 0.177i)9-s + (0.218 + 1.10i)10-s + (0.338 + 1.69i)11-s + (0.277 − 1.39i)12-s + (−0.764 − 0.644i)13-s + (0.786 − 0.525i)14-s + (0.240 − 0.579i)15-s − 0.0783i·16-s + (−0.845 + 0.533i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 221,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28247 + 0.444094i\)
\(L(\frac12)\) \(\approx\) \(2.28247 + 0.444094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.75 + 2.32i)T \)
17 \( 1 + (3.48 - 2.19i)T \)
good2 \( 1 + (-2.09 - 0.869i)T + (1.41 + 1.41i)T^{2} \)
3 \( 1 + (0.864 + 1.29i)T + (-1.14 + 2.77i)T^{2} \)
5 \( 1 + (-0.867 - 1.29i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (-0.864 + 1.29i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (-1.12 - 5.63i)T + (-10.1 + 4.20i)T^{2} \)
19 \( 1 + (4.09 + 1.69i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.37 - 2.25i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (5.78 + 1.14i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (0.0898 - 0.451i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-0.253 + 1.27i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (0.718 + 0.480i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.398 + 0.165i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 4.53iT - 47T^{2} \)
53 \( 1 + (3.43 + 8.29i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-5.34 - 12.9i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (2.17 - 0.433i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (-3.48 - 3.48i)T + 67iT^{2} \)
71 \( 1 + (-9.87 - 1.96i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-12.2 + 8.20i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (-7.10 - 4.74i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-4.95 + 11.9i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + 3.26T + 89T^{2} \)
97 \( 1 + (-1.45 + 0.973i)T + (37.1 - 89.6i)T^{2} \)
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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

−12.77134345861349079205770696715, −11.82762554569081686486091743690, −10.72091927414910343685645904576, −9.530758809661038820454565245422, −7.56607687949986357105954554425, −6.93958517010412985923730703298, −6.29032929527533947862838184174, −5.01846129503515921915984969033, −4.02979819837796221605365224303, −2.23716833289383075217446856073, 2.17264789029903760428929660236, 3.75565754043188832461754920677, 4.87888821596316762335241241003, 5.41319877001082790515230489937, 6.50395884961923532711830363131, 8.510578463425868877579395491568, 9.479357472086901979492472599372, 11.00987056130951699759541845443, 11.14444157532290416534650880677, 12.24826880165913786686144549443

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