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Properties

Label 2-2268-21.17-c1-0-15
Degree 22
Conductor 22682268
Sign 0.624+0.780i0.624 + 0.780i
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  + (−1.00 − 1.73i)5-s + (−2.64 + 0.0655i)7-s + (4.15 + 2.40i)11-s + 0.998i·13-s + (0.0445 − 0.0772i)17-s + (3.68 − 2.12i)19-s + (−0.839 + 0.484i)23-s + (0.492 − 0.853i)25-s + 3.38i·29-s + (−4.35 − 2.51i)31-s + (2.76 + 4.52i)35-s + (−0.0675 − 0.117i)37-s − 11.2·41-s + 7.33·43-s + (1.76 + 3.06i)47-s + ⋯
L(s)  = 1  + (−0.447 − 0.775i)5-s + (−0.999 + 0.0247i)7-s + (1.25 + 0.723i)11-s + 0.276i·13-s + (0.0108 − 0.0187i)17-s + (0.846 − 0.488i)19-s + (−0.175 + 0.101i)23-s + (0.0985 − 0.170i)25-s + 0.628i·29-s + (−0.782 − 0.451i)31-s + (0.467 + 0.764i)35-s + (−0.0111 − 0.0192i)37-s − 1.75·41-s + 1.11·43-s + (0.257 + 0.446i)47-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=((0.624+0.780i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s+1/2) L(s)=((0.624+0.780i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: 0.624+0.780i0.624 + 0.780i
Analytic conductor: 18.110018.1100
Root analytic conductor: 4.255594.25559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2268(1781,⋅)\chi_{2268} (1781, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2268, ( :1/2), 0.624+0.780i)(2,\ 2268,\ (\ :1/2),\ 0.624 + 0.780i)

Particular Values

L(1)L(1) ≈\approx 1.4088127701.408812770
L(12)L(\frac12) ≈\approx 1.4088127701.408812770
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+(2.64−0.0655i)T 1 + (2.64 - 0.0655i)T
good5 1+(1.00+1.73i)T+(−2.5+4.33i)T2 1 + (1.00 + 1.73i)T + (-2.5 + 4.33i)T^{2}
11 1+(−4.15−2.40i)T+(5.5+9.52i)T2 1 + (-4.15 - 2.40i)T + (5.5 + 9.52i)T^{2}
13 1−0.998iT−13T2 1 - 0.998iT - 13T^{2}
17 1+(−0.0445+0.0772i)T+(−8.5−14.7i)T2 1 + (-0.0445 + 0.0772i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.68+2.12i)T+(9.5−16.4i)T2 1 + (-3.68 + 2.12i)T + (9.5 - 16.4i)T^{2}
23 1+(0.839−0.484i)T+(11.5−19.9i)T2 1 + (0.839 - 0.484i)T + (11.5 - 19.9i)T^{2}
29 1−3.38iT−29T2 1 - 3.38iT - 29T^{2}
31 1+(4.35+2.51i)T+(15.5+26.8i)T2 1 + (4.35 + 2.51i)T + (15.5 + 26.8i)T^{2}
37 1+(0.0675+0.117i)T+(−18.5+32.0i)T2 1 + (0.0675 + 0.117i)T + (-18.5 + 32.0i)T^{2}
41 1+11.2T+41T2 1 + 11.2T + 41T^{2}
43 1−7.33T+43T2 1 - 7.33T + 43T^{2}
47 1+(−1.76−3.06i)T+(−23.5+40.7i)T2 1 + (-1.76 - 3.06i)T + (-23.5 + 40.7i)T^{2}
53 1+(−5.31−3.07i)T+(26.5+45.8i)T2 1 + (-5.31 - 3.07i)T + (26.5 + 45.8i)T^{2}
59 1+(−4.88+8.45i)T+(−29.5−51.0i)T2 1 + (-4.88 + 8.45i)T + (-29.5 - 51.0i)T^{2}
61 1+(−11.2+6.51i)T+(30.5−52.8i)T2 1 + (-11.2 + 6.51i)T + (30.5 - 52.8i)T^{2}
67 1+(−7.57+13.1i)T+(−33.5−58.0i)T2 1 + (-7.57 + 13.1i)T + (-33.5 - 58.0i)T^{2}
71 1−4.56iT−71T2 1 - 4.56iT - 71T^{2}
73 1+(−3.73−2.15i)T+(36.5+63.2i)T2 1 + (-3.73 - 2.15i)T + (36.5 + 63.2i)T^{2}
79 1+(5.94+10.2i)T+(−39.5+68.4i)T2 1 + (5.94 + 10.2i)T + (-39.5 + 68.4i)T^{2}
83 1+3.24T+83T2 1 + 3.24T + 83T^{2}
89 1+(1.06+1.84i)T+(−44.5+77.0i)T2 1 + (1.06 + 1.84i)T + (-44.5 + 77.0i)T^{2}
97 1+1.19iT−97T2 1 + 1.19iT - 97T^{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

−9.071071749210324185026879157932, −8.264550309658191306709920212846, −7.16561775246882083767400147528, −6.77647157651036452415645510698, −5.76469275862083646164697294397, −4.82762595780422409817641529271, −4.01791832561760453220697112515, −3.27948935161328281803935849304, −1.90247432923441240009791610590, −0.63683166301827646020471717463, 0.960047812704914356800512619972, 2.54335292293225370212705110130, 3.63864856513276878745749738299, 3.75919946274638844155760458928, 5.35869433633836208115649483154, 6.10468583590347405759379470439, 6.88589860416238642623425061999, 7.35635899233359762942360021596, 8.471120523076945509270082915443, 9.088724020351578058412230996037

Graph of the ZZ-function along the critical line