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Properties

Label 2-2268-63.41-c1-0-10
Degree 22
Conductor 22682268
Sign 0.630−0.775i0.630 - 0.775i
Analytic cond. 18.110018.1100
Root an. cond. 4.255594.25559
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 2.59i)7-s + (1.5 + 0.866i)13-s − 8.66i·19-s + (2.5 + 4.33i)25-s + (9 + 5.19i)31-s + 37-s + (4 + 6.92i)43-s + (−6.5 + 2.59i)49-s + (−7.5 + 4.33i)61-s + (−5.5 + 9.52i)67-s + 1.73i·73-s + (6.5 + 11.2i)79-s + (−1.5 + 4.33i)91-s + (16.5 − 9.52i)97-s + (16.5 + 9.52i)103-s + ⋯
L(s)  = 1  + (0.188 + 0.981i)7-s + (0.416 + 0.240i)13-s − 1.98i·19-s + (0.5 + 0.866i)25-s + (1.61 + 0.933i)31-s + 0.164·37-s + (0.609 + 1.05i)43-s + (−0.928 + 0.371i)49-s + (−0.960 + 0.554i)61-s + (−0.671 + 1.16i)67-s + 0.202i·73-s + (0.731 + 1.26i)79-s + (−0.157 + 0.453i)91-s + (1.67 − 0.967i)97-s + (1.62 + 0.938i)103-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=((0.630−0.775i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s+1/2) L(s)=((0.630−0.775i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: 0.630−0.775i0.630 - 0.775i
Analytic conductor: 18.110018.1100
Root analytic conductor: 4.255594.25559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2268(377,⋅)\chi_{2268} (377, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2268, ( :1/2), 0.630−0.775i)(2,\ 2268,\ (\ :1/2),\ 0.630 - 0.775i)

Particular Values

L(1)L(1) ≈\approx 1.8009302281.800930228
L(12)L(\frac12) ≈\approx 1.8009302281.800930228
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
3 1 1
7 1+(−0.5−2.59i)T 1 + (-0.5 - 2.59i)T
good5 1+(−2.5−4.33i)T2 1 + (-2.5 - 4.33i)T^{2}
11 1+(5.5−9.52i)T2 1 + (5.5 - 9.52i)T^{2}
13 1+(−1.5−0.866i)T+(6.5+11.2i)T2 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2}
17 1+17T2 1 + 17T^{2}
19 1+8.66iT−19T2 1 + 8.66iT - 19T^{2}
23 1+(11.5+19.9i)T2 1 + (11.5 + 19.9i)T^{2}
29 1+(14.5−25.1i)T2 1 + (14.5 - 25.1i)T^{2}
31 1+(−9−5.19i)T+(15.5+26.8i)T2 1 + (-9 - 5.19i)T + (15.5 + 26.8i)T^{2}
37 1−T+37T2 1 - T + 37T^{2}
41 1+(−20.5−35.5i)T2 1 + (-20.5 - 35.5i)T^{2}
43 1+(−4−6.92i)T+(−21.5+37.2i)T2 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2}
47 1+(−23.5+40.7i)T2 1 + (-23.5 + 40.7i)T^{2}
53 1−53T2 1 - 53T^{2}
59 1+(−29.5−51.0i)T2 1 + (-29.5 - 51.0i)T^{2}
61 1+(7.5−4.33i)T+(30.5−52.8i)T2 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2}
67 1+(5.5−9.52i)T+(−33.5−58.0i)T2 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2}
71 1−71T2 1 - 71T^{2}
73 1−1.73iT−73T2 1 - 1.73iT - 73T^{2}
79 1+(−6.5−11.2i)T+(−39.5+68.4i)T2 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2}
83 1+(−41.5+71.8i)T2 1 + (-41.5 + 71.8i)T^{2}
89 1+89T2 1 + 89T^{2}
97 1+(−16.5+9.52i)T+(48.5−84.0i)T2 1 + (-16.5 + 9.52i)T + (48.5 - 84.0i)T^{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

−8.957845594041777441685976503606, −8.614880188907827938813873798234, −7.58884737148203376123949510851, −6.74191425909987842746220278727, −6.05344493236716187471932821006, −5.07074140562705290133021837929, −4.50837833703573721072753568677, −3.14281811639296592433677437549, −2.46893187669859520433949196678, −1.13158635465035578929384526109, 0.73437428858594037551216184151, 1.90042347101419496517922453147, 3.23142853138586706696554153105, 4.04414834191013381453118875332, 4.77873509028592972501557530030, 5.95103662025693935911652799337, 6.45475002832406033965570038753, 7.60180260246159955692598488342, 7.977926946483047116697374987659, 8.816461954399482816526542109352

Graph of the ZZ-function along the critical line