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Properties

Label 2-55-55.14-c1-0-1
Degree 22
Conductor 5555
Sign 0.834−0.550i0.834 - 0.550i
Analytic cond. 0.4391770.439177
Root an. cond. 0.6627040.662704
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  + (0.725 + 0.998i)2-s + (0.346 − 0.112i)3-s + (0.147 − 0.453i)4-s + (−2.11 + 0.713i)5-s + (0.363 + 0.264i)6-s + (−2.45 − 0.798i)7-s + (2.90 − 0.944i)8-s + (−2.31 + 1.68i)9-s + (−2.24 − 1.59i)10-s + (3.12 + 1.12i)11-s − 0.173i·12-s + (−1.62 − 2.23i)13-s + (−0.985 − 3.03i)14-s + (−0.653 + 0.485i)15-s + (2.27 + 1.65i)16-s + (2.26 − 3.11i)17-s + ⋯
L(s)  = 1  + (0.512 + 0.705i)2-s + (0.199 − 0.0649i)3-s + (0.0737 − 0.226i)4-s + (−0.947 + 0.319i)5-s + (0.148 + 0.107i)6-s + (−0.929 − 0.301i)7-s + (1.02 − 0.333i)8-s + (−0.773 + 0.561i)9-s + (−0.711 − 0.505i)10-s + (0.940 + 0.339i)11-s − 0.0501i·12-s + (−0.449 − 0.619i)13-s + (−0.263 − 0.810i)14-s + (−0.168 + 0.125i)15-s + (0.569 + 0.414i)16-s + (0.549 − 0.755i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.834−0.550i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=((0.834−0.550i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.834−0.550i0.834 - 0.550i
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ55(14,⋅)\chi_{55} (14, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1/2), 0.834−0.550i)(2,\ 55,\ (\ :1/2),\ 0.834 - 0.550i)

Particular Values

L(1)L(1) ≈\approx 0.963540+0.289361i0.963540 + 0.289361i
L(12)L(\frac12) ≈\approx 0.963540+0.289361i0.963540 + 0.289361i
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)
bad5 1+(2.11−0.713i)T 1 + (2.11 - 0.713i)T
11 1+(−3.12−1.12i)T 1 + (-3.12 - 1.12i)T
good2 1+(−0.725−0.998i)T+(−0.618+1.90i)T2 1 + (-0.725 - 0.998i)T + (-0.618 + 1.90i)T^{2}
3 1+(−0.346+0.112i)T+(2.42−1.76i)T2 1 + (-0.346 + 0.112i)T + (2.42 - 1.76i)T^{2}
7 1+(2.45+0.798i)T+(5.66+4.11i)T2 1 + (2.45 + 0.798i)T + (5.66 + 4.11i)T^{2}
13 1+(1.62+2.23i)T+(−4.01+12.3i)T2 1 + (1.62 + 2.23i)T + (-4.01 + 12.3i)T^{2}
17 1+(−2.26+3.11i)T+(−5.25−16.1i)T2 1 + (-2.26 + 3.11i)T + (-5.25 - 16.1i)T^{2}
19 1+(−0.0857−0.264i)T+(−15.3+11.1i)T2 1 + (-0.0857 - 0.264i)T + (-15.3 + 11.1i)T^{2}
23 1−8.40iT−23T2 1 - 8.40iT - 23T^{2}
29 1+(1.02−3.16i)T+(−23.4−17.0i)T2 1 + (1.02 - 3.16i)T + (-23.4 - 17.0i)T^{2}
31 1+(0.456−0.331i)T+(9.57−29.4i)T2 1 + (0.456 - 0.331i)T + (9.57 - 29.4i)T^{2}
37 1+(0.497+0.161i)T+(29.9+21.7i)T2 1 + (0.497 + 0.161i)T + (29.9 + 21.7i)T^{2}
41 1+(1.57+4.86i)T+(−33.1+24.0i)T2 1 + (1.57 + 4.86i)T + (-33.1 + 24.0i)T^{2}
43 1+2.54iT−43T2 1 + 2.54iT - 43T^{2}
47 1+(−4.68+1.52i)T+(38.0−27.6i)T2 1 + (-4.68 + 1.52i)T + (38.0 - 27.6i)T^{2}
53 1+(5.12+7.05i)T+(−16.3+50.4i)T2 1 + (5.12 + 7.05i)T + (-16.3 + 50.4i)T^{2}
59 1+(2.31−7.13i)T+(−47.7−34.6i)T2 1 + (2.31 - 7.13i)T + (-47.7 - 34.6i)T^{2}
61 1+(11.4+8.33i)T+(18.8+58.0i)T2 1 + (11.4 + 8.33i)T + (18.8 + 58.0i)T^{2}
67 1−3.20iT−67T2 1 - 3.20iT - 67T^{2}
71 1+(−6.79−4.93i)T+(21.9+67.5i)T2 1 + (-6.79 - 4.93i)T + (21.9 + 67.5i)T^{2}
73 1+(−12.3−4.02i)T+(59.0+42.9i)T2 1 + (-12.3 - 4.02i)T + (59.0 + 42.9i)T^{2}
79 1+(−7.85+5.70i)T+(24.4−75.1i)T2 1 + (-7.85 + 5.70i)T + (24.4 - 75.1i)T^{2}
83 1+(1.93−2.66i)T+(−25.6−78.9i)T2 1 + (1.93 - 2.66i)T + (-25.6 - 78.9i)T^{2}
89 1+2.48T+89T2 1 + 2.48T + 89T^{2}
97 1+(−6.40−8.81i)T+(−29.9+92.2i)T2 1 + (-6.40 - 8.81i)T + (-29.9 + 92.2i)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

−15.36620017734233547355412590916, −14.43552992250599022763929400483, −13.56455870341471929100461587525, −12.14333377275171305939971425018, −10.88740170675470293863057144100, −9.557034928643488224255017332771, −7.73284852000233910814030802180, −6.85672270016418701661153164018, −5.32204466462974643831084745307, −3.52882912728503075106484871815, 3.10293891013009170619354558803, 4.24213330381491402305082889604, 6.45145449956813619574853299824, 8.132153788753637135653223743089, 9.290805876867923238838911064408, 11.01591494732306672212269756143, 12.11400552689035190602793752880, 12.51357000971853668368735529045, 14.02260878519013499332393962570, 15.06108015571473162402207817091

Graph of the ZZ-function along the critical line