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Properties

Label 2-220-20.3-c1-0-14
Degree 22
Conductor 220220
Sign −0.850+0.525i-0.850 + 0.525i
Analytic cond. 1.756701.75670
Root an. cond. 1.325401.32540
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

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

Λ(s)=(220s/2 ΓC(s) L(s)=((−0.850+0.525i) Λ‾(2−s)\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}
Λ(s)=(220s/2 ΓC(s+1/2) L(s)=((−0.850+0.525i) Λ‾(1−s)\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: 22
Conductor: 220220    =    22⋅5⋅112^{2} \cdot 5 \cdot 11
Sign: −0.850+0.525i-0.850 + 0.525i
Analytic conductor: 1.756701.75670
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ220(23,⋅)\chi_{220} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 220, ( :1/2), −0.850+0.525i)(2,\ 220,\ (\ :1/2),\ -0.850 + 0.525i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1+i)T 1 + (1 + i)T
5 1+(1−2i)T 1 + (1 - 2i)T
11 1−iT 1 - iT
good3 1+(2−2i)T−3iT2 1 + (2 - 2i)T - 3iT^{2}
7 1+(1+i)T+7iT2 1 + (1 + i)T + 7iT^{2}
13 1+(4+4i)T+13iT2 1 + (4 + 4i)T + 13iT^{2}
17 1+(−2+2i)T−17iT2 1 + (-2 + 2i)T - 17iT^{2}
19 1−2T+19T2 1 - 2T + 19T^{2}
23 1+(4−4i)T−23iT2 1 + (4 - 4i)T - 23iT^{2}
29 1−2iT−29T2 1 - 2iT - 29T^{2}
31 1+8iT−31T2 1 + 8iT - 31T^{2}
37 1+(5−5i)T−37iT2 1 + (5 - 5i)T - 37iT^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1+(−1+i)T−43iT2 1 + (-1 + i)T - 43iT^{2}
47 1+(4+4i)T+47iT2 1 + (4 + 4i)T + 47iT^{2}
53 1+(−9−9i)T+53iT2 1 + (-9 - 9i)T + 53iT^{2}
59 1+8T+59T2 1 + 8T + 59T^{2}
61 1+10T+61T2 1 + 10T + 61T^{2}
67 1+67iT2 1 + 67iT^{2}
71 1−12iT−71T2 1 - 12iT - 71T^{2}
73 1+(−2−2i)T+73iT2 1 + (-2 - 2i)T + 73iT^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1+(−1+i)T−83iT2 1 + (-1 + i)T - 83iT^{2}
89 1−6iT−89T2 1 - 6iT - 89T^{2}
97 1+(3−3i)T−97iT2 1 + (3 - 3i)T - 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−11.74232555403500297799651086096, −10.71756217274233987093872303941, −10.06476964818689916668247879107, −9.656433953158384843371413337775, −7.85972654731057360047242333339, −6.95985893794383838459495489488, −5.44283590408288886221109332831, −4.10408770385390336147271368246, −3.05343803685162763025309296745, 0, 1.65791358384701540528400149726, 4.77350202420320666474275227815, 5.73192741945282116023582038379, 6.67731876598189408121292457164, 7.54182331162429303070473363800, 8.526512738328720310933327079795, 9.597462312647546564069275835363, 10.82321017671691156828651678220, 12.08468361701946296351947901692

Graph of the ZZ-function along the critical line