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Properties

Label 2-220-20.3-c1-0-2
Degree $2$
Conductor $220$
Sign $0.799 - 0.600i$
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  + (−0.354 − 1.36i)2-s + (−1.16 + 1.16i)3-s + (−1.74 + 0.971i)4-s + (−1.42 − 1.72i)5-s + (2.00 + 1.18i)6-s + (1.60 + 1.60i)7-s + (1.94 + 2.04i)8-s + 0.282i·9-s + (−1.85 + 2.56i)10-s + i·11-s + (0.906 − 3.17i)12-s + (3.94 + 3.94i)13-s + (1.62 − 2.76i)14-s + (3.66 + 0.350i)15-s + (2.11 − 3.39i)16-s + (−4.86 + 4.86i)17-s + ⋯
L(s)  = 1  + (−0.250 − 0.968i)2-s + (−0.673 + 0.673i)3-s + (−0.874 + 0.485i)4-s + (−0.636 − 0.771i)5-s + (0.820 + 0.482i)6-s + (0.605 + 0.605i)7-s + (0.689 + 0.724i)8-s + 0.0940i·9-s + (−0.586 + 0.809i)10-s + 0.301i·11-s + (0.261 − 0.915i)12-s + (1.09 + 1.09i)13-s + (0.434 − 0.737i)14-s + (0.947 + 0.0903i)15-s + (0.528 − 0.848i)16-s + (−1.17 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.799 - 0.600i$
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: \((2,\ 220,\ (\ :1/2),\ 0.799 - 0.600i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637662 + 0.212639i\)
\(L(\frac12)\) \(\approx\) \(0.637662 + 0.212639i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.354 + 1.36i)T \)
5 \( 1 + (1.42 + 1.72i)T \)
11 \( 1 - iT \)
good3 \( 1 + (1.16 - 1.16i)T - 3iT^{2} \)
7 \( 1 + (-1.60 - 1.60i)T + 7iT^{2} \)
13 \( 1 + (-3.94 - 3.94i)T + 13iT^{2} \)
17 \( 1 + (4.86 - 4.86i)T - 17iT^{2} \)
19 \( 1 - 5.54T + 19T^{2} \)
23 \( 1 + (3.01 - 3.01i)T - 23iT^{2} \)
29 \( 1 + 6.65iT - 29T^{2} \)
31 \( 1 - 0.619iT - 31T^{2} \)
37 \( 1 + (-0.294 + 0.294i)T - 37iT^{2} \)
41 \( 1 + 2.27T + 41T^{2} \)
43 \( 1 + (2.52 - 2.52i)T - 43iT^{2} \)
47 \( 1 + (-8.42 - 8.42i)T + 47iT^{2} \)
53 \( 1 + (-2.22 - 2.22i)T + 53iT^{2} \)
59 \( 1 + 1.02T + 59T^{2} \)
61 \( 1 + 4.52T + 61T^{2} \)
67 \( 1 + (3.45 + 3.45i)T + 67iT^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (-3.57 - 3.57i)T + 73iT^{2} \)
79 \( 1 - 2.91T + 79T^{2} \)
83 \( 1 + (-9.90 + 9.90i)T - 83iT^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 + (0.328 - 0.328i)T - 97iT^{2} \)
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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

−11.82509622980105370575464985428, −11.56612355178822024037591133195, −10.71474422176355619871836919019, −9.512860809632527566546148785680, −8.696993599214977583794867206143, −7.81737852667245224351661456473, −5.78314296886247503140283202954, −4.62614042031153995175681163074, −3.96275256325090333761856762225, −1.76171195040172244662055396504, 0.72430699420033468704896578392, 3.62492579445557852930175202565, 5.13005292666990074950801979224, 6.30005004215301743199899314779, 7.11698473101565287176271336495, 7.83369952270886482603732917710, 8.937389007123087755887957409451, 10.45280750523032070525086321214, 11.12737794653577316907660306753, 12.11786042130547443126178139645

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