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Properties

Label 2-50-25.6-c1-0-1
Degree 22
Conductor 5050
Sign 0.890−0.455i0.890 - 0.455i
Analytic cond. 0.3992520.399252
Root an. cond. 0.6318630.631863
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  + (0.309 + 0.951i)2-s + (1.09 − 0.792i)3-s + (−0.809 + 0.587i)4-s + (−2.15 − 0.587i)5-s + (1.09 + 0.792i)6-s − 0.833·7-s + (−0.809 − 0.587i)8-s + (−0.365 + 1.12i)9-s + (−0.107 − 2.23i)10-s + (0.257 + 0.792i)11-s + (−0.416 + 1.28i)12-s + (1.41 − 4.34i)13-s + (−0.257 − 0.792i)14-s + (−2.81 + 1.06i)15-s + (0.309 − 0.951i)16-s + (−4.41 − 3.20i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.629 − 0.457i)3-s + (−0.404 + 0.293i)4-s + (−0.964 − 0.262i)5-s + (0.445 + 0.323i)6-s − 0.314·7-s + (−0.286 − 0.207i)8-s + (−0.121 + 0.374i)9-s + (−0.0340 − 0.706i)10-s + (0.0776 + 0.238i)11-s + (−0.120 + 0.370i)12-s + (0.391 − 1.20i)13-s + (−0.0688 − 0.211i)14-s + (−0.727 + 0.275i)15-s + (0.0772 − 0.237i)16-s + (−1.06 − 0.777i)17-s + ⋯

Functional equation

Λ(s)=(50s/2 ΓC(s) L(s)=((0.890−0.455i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(50s/2 ΓC(s+1/2) L(s)=((0.890−0.455i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5050    =    2⋅522 \cdot 5^{2}
Sign: 0.890−0.455i0.890 - 0.455i
Analytic conductor: 0.3992520.399252
Root analytic conductor: 0.6318630.631863
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ50(31,⋅)\chi_{50} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 50, ( :1/2), 0.890−0.455i)(2,\ 50,\ (\ :1/2),\ 0.890 - 0.455i)

Particular Values

L(1)L(1) ≈\approx 0.884801+0.213399i0.884801 + 0.213399i
L(12)L(\frac12) ≈\approx 0.884801+0.213399i0.884801 + 0.213399i
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+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
5 1+(2.15+0.587i)T 1 + (2.15 + 0.587i)T
good3 1+(−1.09+0.792i)T+(0.927−2.85i)T2 1 + (-1.09 + 0.792i)T + (0.927 - 2.85i)T^{2}
7 1+0.833T+7T2 1 + 0.833T + 7T^{2}
11 1+(−0.257−0.792i)T+(−8.89+6.46i)T2 1 + (-0.257 - 0.792i)T + (-8.89 + 6.46i)T^{2}
13 1+(−1.41+4.34i)T+(−10.5−7.64i)T2 1 + (-1.41 + 4.34i)T + (-10.5 - 7.64i)T^{2}
17 1+(4.41+3.20i)T+(5.25+16.1i)T2 1 + (4.41 + 3.20i)T + (5.25 + 16.1i)T^{2}
19 1+(−7.00−5.08i)T+(5.87+18.0i)T2 1 + (-7.00 - 5.08i)T + (5.87 + 18.0i)T^{2}
23 1+(−1.09−3.35i)T+(−18.6+13.5i)T2 1 + (-1.09 - 3.35i)T + (-18.6 + 13.5i)T^{2}
29 1+(2.64−1.92i)T+(8.96−27.5i)T2 1 + (2.64 - 1.92i)T + (8.96 - 27.5i)T^{2}
31 1+(4.85+3.52i)T+(9.57+29.4i)T2 1 + (4.85 + 3.52i)T + (9.57 + 29.4i)T^{2}
37 1+(−2.26+6.95i)T+(−29.9−21.7i)T2 1 + (-2.26 + 6.95i)T + (-29.9 - 21.7i)T^{2}
41 1+(−0.576+1.77i)T+(−33.1−24.0i)T2 1 + (-0.576 + 1.77i)T + (-33.1 - 24.0i)T^{2}
43 1+1.63T+43T2 1 + 1.63T + 43T^{2}
47 1+(−0.674+0.489i)T+(14.5−44.6i)T2 1 + (-0.674 + 0.489i)T + (14.5 - 44.6i)T^{2}
53 1+(5.19−3.77i)T+(16.3−50.4i)T2 1 + (5.19 - 3.77i)T + (16.3 - 50.4i)T^{2}
59 1+(−4.18+12.8i)T+(−47.7−34.6i)T2 1 + (-4.18 + 12.8i)T + (-47.7 - 34.6i)T^{2}
61 1+(−1.81−5.59i)T+(−49.3+35.8i)T2 1 + (-1.81 - 5.59i)T + (-49.3 + 35.8i)T^{2}
67 1+(1.21+0.881i)T+(20.7+63.7i)T2 1 + (1.21 + 0.881i)T + (20.7 + 63.7i)T^{2}
71 1+(−1.91+1.38i)T+(21.9−67.5i)T2 1 + (-1.91 + 1.38i)T + (21.9 - 67.5i)T^{2}
73 1+(1.02+3.16i)T+(−59.0+42.9i)T2 1 + (1.02 + 3.16i)T + (-59.0 + 42.9i)T^{2}
79 1+(4.18−3.03i)T+(24.4−75.1i)T2 1 + (4.18 - 3.03i)T + (24.4 - 75.1i)T^{2}
83 1+(−9.97−7.25i)T+(25.6+78.9i)T2 1 + (-9.97 - 7.25i)T + (25.6 + 78.9i)T^{2}
89 1+(2.16+6.66i)T+(−72.0+52.3i)T2 1 + (2.16 + 6.66i)T + (-72.0 + 52.3i)T^{2}
97 1+(8.97−6.51i)T+(29.9−92.2i)T2 1 + (8.97 - 6.51i)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

−15.71722160923512237471653514967, −14.59849972747249108909689432482, −13.45810619216296466924212719357, −12.60639995224576316036787102613, −11.19477810397215719145609273934, −9.299343603771264096174189573705, −7.995955762103588709137229109224, −7.31881562226818636948363357107, −5.34535994999870021536395595042, −3.44266246576126949283524513030, 3.17459120572987956606991684069, 4.36420157987861449824327394935, 6.71674076957440159812442047772, 8.573252849835968570618815211044, 9.475582019359727795610839772923, 11.04574701177117349738944016356, 11.83181206204205240403486297782, 13.26576119433431699757004728734, 14.40079464679263799472507203816, 15.34091117881981262133671140841

Graph of the ZZ-function along the critical line