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Properties

Label 2-55-11.6-c2-0-4
Degree 22
Conductor 5555
Sign 0.994−0.108i0.994 - 0.108i
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  + (1.57 − 0.512i)2-s + (2.41 + 1.75i)3-s + (−1.01 + 0.737i)4-s + (0.690 − 2.12i)5-s + (4.70 + 1.52i)6-s + (−1.31 − 1.80i)7-s + (−5.11 + 7.04i)8-s + (−0.0320 − 0.0986i)9-s − 3.70i·10-s + (−0.949 − 10.9i)11-s − 3.74·12-s + (−1.75 + 0.568i)13-s + (−2.98 − 2.17i)14-s + (5.39 − 3.92i)15-s + (−2.90 + 8.94i)16-s + (7.97 + 2.59i)17-s + ⋯
L(s)  = 1  + (0.787 − 0.256i)2-s + (0.804 + 0.584i)3-s + (−0.253 + 0.184i)4-s + (0.138 − 0.425i)5-s + (0.783 + 0.254i)6-s + (−0.187 − 0.257i)7-s + (−0.639 + 0.880i)8-s + (−0.00356 − 0.0109i)9-s − 0.370i·10-s + (−0.0862 − 0.996i)11-s − 0.311·12-s + (−0.134 + 0.0437i)13-s + (−0.213 − 0.155i)14-s + (0.359 − 0.261i)15-s + (−0.181 + 0.559i)16-s + (0.469 + 0.152i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.994−0.108i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1) L(s)=((0.994−0.108i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.994−0.108i0.994 - 0.108i
Analytic conductor: 1.498641.49864
Root analytic conductor: 1.224191.22419
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ55(6,⋅)\chi_{55} (6, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1), 0.994−0.108i)(2,\ 55,\ (\ :1),\ 0.994 - 0.108i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.79124+0.0971539i1.79124 + 0.0971539i
L(12)L(\frac12) ≈\approx 1.79124+0.0971539i1.79124 + 0.0971539i
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+(−0.690+2.12i)T 1 + (-0.690 + 2.12i)T
11 1+(0.949+10.9i)T 1 + (0.949 + 10.9i)T
good2 1+(−1.57+0.512i)T+(3.23−2.35i)T2 1 + (-1.57 + 0.512i)T + (3.23 - 2.35i)T^{2}
3 1+(−2.41−1.75i)T+(2.78+8.55i)T2 1 + (-2.41 - 1.75i)T + (2.78 + 8.55i)T^{2}
7 1+(1.31+1.80i)T+(−15.1+46.6i)T2 1 + (1.31 + 1.80i)T + (-15.1 + 46.6i)T^{2}
13 1+(1.75−0.568i)T+(136.−99.3i)T2 1 + (1.75 - 0.568i)T + (136. - 99.3i)T^{2}
17 1+(−7.97−2.59i)T+(233.+169.i)T2 1 + (-7.97 - 2.59i)T + (233. + 169. i)T^{2}
19 1+(15.6−21.4i)T+(−111.−343.i)T2 1 + (15.6 - 21.4i)T + (-111. - 343. i)T^{2}
23 1+7.63T+529T2 1 + 7.63T + 529T^{2}
29 1+(−31.8−43.7i)T+(−259.+799.i)T2 1 + (-31.8 - 43.7i)T + (-259. + 799. i)T^{2}
31 1+(−1.08−3.32i)T+(−777.+564.i)T2 1 + (-1.08 - 3.32i)T + (-777. + 564. i)T^{2}
37 1+(−37.8+27.5i)T+(423.−1.30e3i)T2 1 + (-37.8 + 27.5i)T + (423. - 1.30e3i)T^{2}
41 1+(15.9−21.9i)T+(−519.−1.59e3i)T2 1 + (15.9 - 21.9i)T + (-519. - 1.59e3i)T^{2}
43 1+14.9iT−1.84e3T2 1 + 14.9iT - 1.84e3T^{2}
47 1+(−58.0−42.1i)T+(682.+2.10e3i)T2 1 + (-58.0 - 42.1i)T + (682. + 2.10e3i)T^{2}
53 1+(12.1+37.4i)T+(−2.27e3+1.65e3i)T2 1 + (12.1 + 37.4i)T + (-2.27e3 + 1.65e3i)T^{2}
59 1+(29.1−21.1i)T+(1.07e3−3.31e3i)T2 1 + (29.1 - 21.1i)T + (1.07e3 - 3.31e3i)T^{2}
61 1+(44.0+14.3i)T+(3.01e3+2.18e3i)T2 1 + (44.0 + 14.3i)T + (3.01e3 + 2.18e3i)T^{2}
67 1+60.8T+4.48e3T2 1 + 60.8T + 4.48e3T^{2}
71 1+(37.8−116.i)T+(−4.07e3−2.96e3i)T2 1 + (37.8 - 116. i)T + (-4.07e3 - 2.96e3i)T^{2}
73 1+(35.1+48.3i)T+(−1.64e3+5.06e3i)T2 1 + (35.1 + 48.3i)T + (-1.64e3 + 5.06e3i)T^{2}
79 1+(81.5−26.4i)T+(5.04e3−3.66e3i)T2 1 + (81.5 - 26.4i)T + (5.04e3 - 3.66e3i)T^{2}
83 1+(43.6+14.1i)T+(5.57e3+4.04e3i)T2 1 + (43.6 + 14.1i)T + (5.57e3 + 4.04e3i)T^{2}
89 1−121.T+7.92e3T2 1 - 121.T + 7.92e3T^{2}
97 1+(10.2+31.6i)T+(−7.61e3+5.53e3i)T2 1 + (10.2 + 31.6i)T + (-7.61e3 + 5.53e3i)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

−14.61666715793757846189910194431, −14.10254890169820880273709418083, −12.97397489144528228532130332024, −11.99582292172056680324244034249, −10.38832094637274933134460001664, −9.016529936706516464057414141705, −8.213706890221770812603474180236, −5.87139879498671209113791105452, −4.26180830324559000391711888376, −3.14510052694506305719347208199, 2.64499136994518620688621738412, 4.58803548678198688954053132490, 6.25408955639800966496593171888, 7.53966781056248200707067952511, 9.043871941501991432257292568294, 10.23936504750582545333141117332, 12.10466849685415887816061351272, 13.17570266475333087106819317574, 13.86917624051539787446674601829, 14.86651484620473640732395204098

Graph of the ZZ-function along the critical line