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Properties

Label 2-220-20.3-c1-0-23
Degree $2$
Conductor $220$
Sign $-0.851 + 0.525i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
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  + (−0.777 − 1.18i)2-s + (0.810 − 0.810i)3-s + (−0.792 + 1.83i)4-s + (0.127 − 2.23i)5-s + (−1.58 − 0.327i)6-s + (−2.32 − 2.32i)7-s + (2.78 − 0.491i)8-s + 1.68i·9-s + (−2.73 + 1.58i)10-s − i·11-s + (0.846 + 2.13i)12-s + (−2.59 − 2.59i)13-s + (−0.941 + 4.55i)14-s + (−1.70 − 1.91i)15-s + (−2.74 − 2.90i)16-s + (3.45 − 3.45i)17-s + ⋯
L(s)  = 1  + (−0.549 − 0.835i)2-s + (0.468 − 0.468i)3-s + (−0.396 + 0.918i)4-s + (0.0571 − 0.998i)5-s + (−0.648 − 0.133i)6-s + (−0.879 − 0.879i)7-s + (0.984 − 0.173i)8-s + 0.561i·9-s + (−0.865 + 0.500i)10-s − 0.301i·11-s + (0.244 + 0.615i)12-s + (−0.719 − 0.719i)13-s + (−0.251 + 1.21i)14-s + (−0.440 − 0.494i)15-s + (−0.686 − 0.727i)16-s + (0.838 − 0.838i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $-0.851 + 0.525i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ -0.851 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.240259 - 0.846975i\)
\(L(\frac12)\) \(\approx\) \(0.240259 - 0.846975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.777 + 1.18i)T \)
5 \( 1 + (-0.127 + 2.23i)T \)
11 \( 1 + iT \)
good3 \( 1 + (-0.810 + 0.810i)T - 3iT^{2} \)
7 \( 1 + (2.32 + 2.32i)T + 7iT^{2} \)
13 \( 1 + (2.59 + 2.59i)T + 13iT^{2} \)
17 \( 1 + (-3.45 + 3.45i)T - 17iT^{2} \)
19 \( 1 - 2.29T + 19T^{2} \)
23 \( 1 + (4.20 - 4.20i)T - 23iT^{2} \)
29 \( 1 + 0.133iT - 29T^{2} \)
31 \( 1 - 1.32iT - 31T^{2} \)
37 \( 1 + (-7.23 + 7.23i)T - 37iT^{2} \)
41 \( 1 - 7.76T + 41T^{2} \)
43 \( 1 + (-5.00 + 5.00i)T - 43iT^{2} \)
47 \( 1 + (-6.17 - 6.17i)T + 47iT^{2} \)
53 \( 1 + (-3.93 - 3.93i)T + 53iT^{2} \)
59 \( 1 - 6.54T + 59T^{2} \)
61 \( 1 + 7.15T + 61T^{2} \)
67 \( 1 + (-3.80 - 3.80i)T + 67iT^{2} \)
71 \( 1 + 12.5iT - 71T^{2} \)
73 \( 1 + (1.45 + 1.45i)T + 73iT^{2} \)
79 \( 1 - 2.50T + 79T^{2} \)
83 \( 1 + (6.44 - 6.44i)T - 83iT^{2} \)
89 \( 1 - 14.4iT - 89T^{2} \)
97 \( 1 + (9.18 - 9.18i)T - 97iT^{2} \)
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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

−12.09587789920623482739721417449, −10.79248725306069002374272117489, −9.812989521242164563458656763662, −9.173069258255759991479144314092, −7.77342430976627791919485362814, −7.50884900959397383473480387410, −5.42861891712000169864901941946, −3.97977599135669012664777718950, −2.64939106823121948833951909983, −0.869357512258204667325669050461, 2.60819504366570245911877353731, 4.11390058490478353458082899614, 5.87256625714373589708417826754, 6.55082366409102869205668320852, 7.68202859039474293546511540095, 8.868635697147048428151223468887, 9.787924219237480963933614504334, 10.11544773118741693697921881683, 11.64431481599743063442738455024, 12.71208952866861696742549240364

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