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Properties

Label 2-220-20.7-c1-0-25
Degree $2$
Conductor $220$
Sign $0.850 + 0.525i$
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  + (1 − i)2-s + (2 + 2i)3-s − 2i·4-s + (−1 − 2i)5-s + 4·6-s + (1 − i)7-s + (−2 − 2i)8-s + 5i·9-s + (−3 − i)10-s + i·11-s + (4 − 4i)12-s + (−4 + 4i)13-s − 2i·14-s + (2 − 6i)15-s − 4·16-s + (2 + 2i)17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (1.15 + 1.15i)3-s − i·4-s + (−0.447 − 0.894i)5-s + 1.63·6-s + (0.377 − 0.377i)7-s + (−0.707 − 0.707i)8-s + 1.66i·9-s + (−0.948 − 0.316i)10-s + 0.301i·11-s + (1.15 − 1.15i)12-s + (−1.10 + 1.10i)13-s − 0.534i·14-s + (0.516 − 1.54i)15-s − 16-s + (0.485 + 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08965 - 0.593626i\)
\(L(\frac12)\) \(\approx\) \(2.08965 - 0.593626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (1 + 2i)T \)
11 \( 1 - iT \)
good3 \( 1 + (-2 - 2i)T + 3iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
13 \( 1 + (4 - 4i)T - 13iT^{2} \)
17 \( 1 + (-2 - 2i)T + 17iT^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (-4 - 4i)T + 23iT^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 8iT - 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + (1 + i)T + 43iT^{2} \)
47 \( 1 + (-4 + 4i)T - 47iT^{2} \)
53 \( 1 + (-9 + 9i)T - 53iT^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (-2 + 2i)T - 73iT^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + (1 + i)T + 83iT^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + (3 + 3i)T + 97iT^{2} \)
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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

−12.18867249337151837304572447647, −11.29397202068506082252926055128, −10.11648237488714637021480229585, −9.453633220649238396697070769593, −8.624312871136564018272125362135, −7.33597006180653478265023245178, −5.25738493624964375170500868039, −4.38338932882092220423139813221, −3.72139969892751743934681285446, −2.09092810389286353199831228164, 2.59057171889261465725008868322, 3.27808248789426262086213812530, 5.10946208062674339360294620042, 6.61635367314294036585971940962, 7.31245446689933418325187777012, 8.077176421466973270283863362218, 8.831921865954385221617692499110, 10.54137573775509692327993230978, 12.01700346542955690325834525975, 12.45985598901552531534436160613

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