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Properties

Label 2-55-11.7-c2-0-2
Degree 22
Conductor 5555
Sign 0.407−0.913i0.407 - 0.913i
Analytic cond. 1.498641.49864
Root an. cond. 1.224191.22419
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0936 − 0.128i)2-s + (−0.602 + 1.85i)3-s + (1.22 + 3.78i)4-s + (−1.80 + 1.31i)5-s + (0.182 + 0.251i)6-s + (0.0125 − 0.00408i)7-s + (1.20 + 0.392i)8-s + (4.20 + 3.05i)9-s + 0.356i·10-s + (6.77 − 8.66i)11-s − 7.75·12-s + (3.39 − 4.67i)13-s + (0.000651 − 0.00200i)14-s + (−1.34 − 4.14i)15-s + (−12.6 + 9.22i)16-s + (−17.8 − 24.5i)17-s + ⋯
L(s)  = 1  + (0.0468 − 0.0644i)2-s + (−0.200 + 0.618i)3-s + (0.307 + 0.945i)4-s + (−0.361 + 0.262i)5-s + (0.0304 + 0.0419i)6-s + (0.00179 − 0.000583i)7-s + (0.151 + 0.0491i)8-s + (0.466 + 0.339i)9-s + 0.0356i·10-s + (0.616 − 0.787i)11-s − 0.646·12-s + (0.261 − 0.359i)13-s + (4.65e−5 − 0.000143i)14-s + (−0.0898 − 0.276i)15-s + (−0.793 + 0.576i)16-s + (−1.04 − 1.44i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.407−0.913i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1) L(s)=((0.407−0.913i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.407−0.913i0.407 - 0.913i
Analytic conductor: 1.498641.49864
Root analytic conductor: 1.224191.22419
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ55(51,⋅)\chi_{55} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1), 0.407−0.913i)(2,\ 55,\ (\ :1),\ 0.407 - 0.913i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.974991+0.632830i0.974991 + 0.632830i
L(12)L(\frac12) ≈\approx 0.974991+0.632830i0.974991 + 0.632830i
L(2)L(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+(1.80−1.31i)T 1 + (1.80 - 1.31i)T
11 1+(−6.77+8.66i)T 1 + (-6.77 + 8.66i)T
good2 1+(−0.0936+0.128i)T+(−1.23−3.80i)T2 1 + (-0.0936 + 0.128i)T + (-1.23 - 3.80i)T^{2}
3 1+(0.602−1.85i)T+(−7.28−5.29i)T2 1 + (0.602 - 1.85i)T + (-7.28 - 5.29i)T^{2}
7 1+(−0.0125+0.00408i)T+(39.6−28.8i)T2 1 + (-0.0125 + 0.00408i)T + (39.6 - 28.8i)T^{2}
13 1+(−3.39+4.67i)T+(−52.2−160.i)T2 1 + (-3.39 + 4.67i)T + (-52.2 - 160. i)T^{2}
17 1+(17.8+24.5i)T+(−89.3+274.i)T2 1 + (17.8 + 24.5i)T + (-89.3 + 274. i)T^{2}
19 1+(−12.4−4.03i)T+(292.+212.i)T2 1 + (-12.4 - 4.03i)T + (292. + 212. i)T^{2}
23 1−13.0T+529T2 1 - 13.0T + 529T^{2}
29 1+(−27.2+8.86i)T+(680.−494.i)T2 1 + (-27.2 + 8.86i)T + (680. - 494. i)T^{2}
31 1+(−10.2−7.46i)T+(296.+913.i)T2 1 + (-10.2 - 7.46i)T + (296. + 913. i)T^{2}
37 1+(8.14+25.0i)T+(−1.10e3+804.i)T2 1 + (8.14 + 25.0i)T + (-1.10e3 + 804. i)T^{2}
41 1+(−2.87−0.933i)T+(1.35e3+988.i)T2 1 + (-2.87 - 0.933i)T + (1.35e3 + 988. i)T^{2}
43 1−73.0iT−1.84e3T2 1 - 73.0iT - 1.84e3T^{2}
47 1+(−6.67+20.5i)T+(−1.78e3−1.29e3i)T2 1 + (-6.67 + 20.5i)T + (-1.78e3 - 1.29e3i)T^{2}
53 1+(55.8+40.6i)T+(868.+2.67e3i)T2 1 + (55.8 + 40.6i)T + (868. + 2.67e3i)T^{2}
59 1+(18.9+58.2i)T+(−2.81e3+2.04e3i)T2 1 + (18.9 + 58.2i)T + (-2.81e3 + 2.04e3i)T^{2}
61 1+(56.5+77.8i)T+(−1.14e3+3.53e3i)T2 1 + (56.5 + 77.8i)T + (-1.14e3 + 3.53e3i)T^{2}
67 1−67.6T+4.48e3T2 1 - 67.6T + 4.48e3T^{2}
71 1+(41.3−30.0i)T+(1.55e3−4.79e3i)T2 1 + (41.3 - 30.0i)T + (1.55e3 - 4.79e3i)T^{2}
73 1+(41.1−13.3i)T+(4.31e3−3.13e3i)T2 1 + (41.1 - 13.3i)T + (4.31e3 - 3.13e3i)T^{2}
79 1+(4.49−6.18i)T+(−1.92e3−5.93e3i)T2 1 + (4.49 - 6.18i)T + (-1.92e3 - 5.93e3i)T^{2}
83 1+(0.721+0.992i)T+(−2.12e3+6.55e3i)T2 1 + (0.721 + 0.992i)T + (-2.12e3 + 6.55e3i)T^{2}
89 1+117.T+7.92e3T2 1 + 117.T + 7.92e3T^{2}
97 1+(−106.−77.6i)T+(2.90e3+8.94e3i)T2 1 + (-106. - 77.6i)T + (2.90e3 + 8.94e3i)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

−15.85443441187677052845677914486, −14.09542559467638864798423703347, −12.97732716157304661703658042083, −11.60134920690568579360118636308, −10.96170324815448010576794840919, −9.371510093758658077962256888102, −7.991165107587407898105706674005, −6.72388322695164616008034185032, −4.64480615497035529434112458894, −3.18818771425762984152907227252, 1.49639468571285230802312960616, 4.45516816485103377154957278940, 6.26000434721326809175091486122, 7.15008973866106776981374707150, 8.947514828715297823902983132753, 10.24955509158682261816095222362, 11.52932520611932127849014546233, 12.54198766888066049034633809416, 13.72517997110996518955571941978, 15.05168994297607470498430042473

Graph of the ZZ-function along the critical line