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Properties

Label 2-55-55.18-c1-0-3
Degree 22
Conductor 5555
Sign 0.319+0.947i0.319 + 0.947i
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  + (−0.237 − 1.50i)2-s + (0.710 − 0.361i)3-s + (−0.295 + 0.0959i)4-s + (−1.71 + 1.43i)5-s + (−0.712 − 0.980i)6-s + (0.0869 − 0.170i)7-s + (−1.16 − 2.28i)8-s + (−1.38 + 1.91i)9-s + (2.56 + 2.23i)10-s + (1.77 + 2.80i)11-s + (−0.175 + 0.175i)12-s + (3.05 − 0.484i)13-s + (−0.276 − 0.0899i)14-s + (−0.698 + 1.63i)15-s + (−3.66 + 2.65i)16-s + (−3.66 − 0.579i)17-s + ⋯
L(s)  = 1  + (−0.168 − 1.06i)2-s + (0.410 − 0.208i)3-s + (−0.147 + 0.0479i)4-s + (−0.766 + 0.641i)5-s + (−0.290 − 0.400i)6-s + (0.0328 − 0.0644i)7-s + (−0.412 − 0.808i)8-s + (−0.463 + 0.637i)9-s + (0.810 + 0.706i)10-s + (0.535 + 0.844i)11-s + (−0.0505 + 0.0505i)12-s + (0.847 − 0.134i)13-s + (−0.0739 − 0.0240i)14-s + (−0.180 + 0.423i)15-s + (−0.915 + 0.664i)16-s + (−0.887 − 0.140i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.319+0.947i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=((0.319+0.947i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.319+0.947i0.319 + 0.947i
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ55(18,⋅)\chi_{55} (18, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1/2), 0.319+0.947i)(2,\ 55,\ (\ :1/2),\ 0.319 + 0.947i)

Particular Values

L(1)L(1) ≈\approx 0.686378−0.493054i0.686378 - 0.493054i
L(12)L(\frac12) ≈\approx 0.686378−0.493054i0.686378 - 0.493054i
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+(1.71−1.43i)T 1 + (1.71 - 1.43i)T
11 1+(−1.77−2.80i)T 1 + (-1.77 - 2.80i)T
good2 1+(0.237+1.50i)T+(−1.90+0.618i)T2 1 + (0.237 + 1.50i)T + (-1.90 + 0.618i)T^{2}
3 1+(−0.710+0.361i)T+(1.76−2.42i)T2 1 + (-0.710 + 0.361i)T + (1.76 - 2.42i)T^{2}
7 1+(−0.0869+0.170i)T+(−4.11−5.66i)T2 1 + (-0.0869 + 0.170i)T + (-4.11 - 5.66i)T^{2}
13 1+(−3.05+0.484i)T+(12.3−4.01i)T2 1 + (-3.05 + 0.484i)T + (12.3 - 4.01i)T^{2}
17 1+(3.66+0.579i)T+(16.1+5.25i)T2 1 + (3.66 + 0.579i)T + (16.1 + 5.25i)T^{2}
19 1+(−0.229+0.707i)T+(−15.3−11.1i)T2 1 + (-0.229 + 0.707i)T + (-15.3 - 11.1i)T^{2}
23 1+(1.14+1.14i)T+23iT2 1 + (1.14 + 1.14i)T + 23iT^{2}
29 1+(2.95+9.07i)T+(−23.4+17.0i)T2 1 + (2.95 + 9.07i)T + (-23.4 + 17.0i)T^{2}
31 1+(0.283+0.206i)T+(9.57+29.4i)T2 1 + (0.283 + 0.206i)T + (9.57 + 29.4i)T^{2}
37 1+(4.81+2.45i)T+(21.7+29.9i)T2 1 + (4.81 + 2.45i)T + (21.7 + 29.9i)T^{2}
41 1+(−6.36−2.06i)T+(33.1+24.0i)T2 1 + (-6.36 - 2.06i)T + (33.1 + 24.0i)T^{2}
43 1+(3.72−3.72i)T−43iT2 1 + (3.72 - 3.72i)T - 43iT^{2}
47 1+(−5.61−11.0i)T+(−27.6+38.0i)T2 1 + (-5.61 - 11.0i)T + (-27.6 + 38.0i)T^{2}
53 1+(1.41+8.91i)T+(−50.4+16.3i)T2 1 + (1.41 + 8.91i)T + (-50.4 + 16.3i)T^{2}
59 1+(−9.15+2.97i)T+(47.7−34.6i)T2 1 + (-9.15 + 2.97i)T + (47.7 - 34.6i)T^{2}
61 1+(3.46+4.76i)T+(−18.8+58.0i)T2 1 + (3.46 + 4.76i)T + (-18.8 + 58.0i)T^{2}
67 1+(−4.13+4.13i)T−67iT2 1 + (-4.13 + 4.13i)T - 67iT^{2}
71 1+(−9.27+6.73i)T+(21.9−67.5i)T2 1 + (-9.27 + 6.73i)T + (21.9 - 67.5i)T^{2}
73 1+(−2.14−1.09i)T+(42.9+59.0i)T2 1 + (-2.14 - 1.09i)T + (42.9 + 59.0i)T^{2}
79 1+(−0.542−0.394i)T+(24.4+75.1i)T2 1 + (-0.542 - 0.394i)T + (24.4 + 75.1i)T^{2}
83 1+(2.60−16.4i)T+(−78.9−25.6i)T2 1 + (2.60 - 16.4i)T + (-78.9 - 25.6i)T^{2}
89 1−7.92iT−89T2 1 - 7.92iT - 89T^{2}
97 1+(1.36−0.215i)T+(92.2−29.9i)T2 1 + (1.36 - 0.215i)T + (92.2 - 29.9i)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

−15.09797627552240304524223808367, −13.85641317938197823534973907048, −12.61994507292337017696407819410, −11.41684266865311522881569135012, −10.83333663261056399294149800401, −9.412251293715021347171985683569, −7.945566117253026051992231013922, −6.58075031946221013134107698875, −3.98321059091092992012627531975, −2.40033684023239449929250978450, 3.68367733179799525995542556162, 5.66924853502106264560026071016, 7.01841093085232033338575205167, 8.661348894057334204091716597121, 8.768723831794898025084569732681, 11.13552817553577831567012186633, 12.06669704392461125776519582535, 13.68488695239633562581376148841, 14.79082013430984784238804762543, 15.64223678005457971085992101195

Graph of the ZZ-function along the critical line