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Properties

Label 2-850-17.9-c1-0-2
Degree 22
Conductor 850850
Sign 0.0784−0.996i0.0784 - 0.996i
Analytic cond. 6.787286.78728
Root an. cond. 2.605242.60524
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  + (−0.707 + 0.707i)2-s + (−1.36 − 0.564i)3-s − 1.00i·4-s + (1.36 − 0.564i)6-s + (−1.35 − 3.27i)7-s + (0.707 + 0.707i)8-s + (−0.581 − 0.581i)9-s + (−4.35 + 1.80i)11-s + (−0.564 + 1.36i)12-s + 5.47i·13-s + (3.27 + 1.35i)14-s − 1.00·16-s + (−2.52 − 3.25i)17-s + 0.822·18-s + (0.857 − 0.857i)19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.787 − 0.326i)3-s − 0.500i·4-s + (0.556 − 0.230i)6-s + (−0.513 − 1.23i)7-s + (0.250 + 0.250i)8-s + (−0.193 − 0.193i)9-s + (−1.31 + 0.543i)11-s + (−0.163 + 0.393i)12-s + 1.51i·13-s + (0.876 + 0.362i)14-s − 0.250·16-s + (−0.613 − 0.789i)17-s + 0.193·18-s + (0.196 − 0.196i)19-s + ⋯

Functional equation

Λ(s)=(850s/2 ΓC(s) L(s)=((0.0784−0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0784 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(850s/2 ΓC(s+1/2) L(s)=((0.0784−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0784 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 850850    =    2⋅52⋅172 \cdot 5^{2} \cdot 17
Sign: 0.0784−0.996i0.0784 - 0.996i
Analytic conductor: 6.787286.78728
Root analytic conductor: 2.605242.60524
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ850(451,⋅)\chi_{850} (451, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 850, ( :1/2), 0.0784−0.996i)(2,\ 850,\ (\ :1/2),\ 0.0784 - 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.311153+0.287620i0.311153 + 0.287620i
L(12)L(\frac12) ≈\approx 0.311153+0.287620i0.311153 + 0.287620i
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
5 1 1
17 1+(2.52+3.25i)T 1 + (2.52 + 3.25i)T
good3 1+(1.36+0.564i)T+(2.12+2.12i)T2 1 + (1.36 + 0.564i)T + (2.12 + 2.12i)T^{2}
7 1+(1.35+3.27i)T+(−4.94+4.94i)T2 1 + (1.35 + 3.27i)T + (-4.94 + 4.94i)T^{2}
11 1+(4.35−1.80i)T+(7.77−7.77i)T2 1 + (4.35 - 1.80i)T + (7.77 - 7.77i)T^{2}
13 1−5.47iT−13T2 1 - 5.47iT - 13T^{2}
19 1+(−0.857+0.857i)T−19iT2 1 + (-0.857 + 0.857i)T - 19iT^{2}
23 1+(−5.28+2.18i)T+(16.2−16.2i)T2 1 + (-5.28 + 2.18i)T + (16.2 - 16.2i)T^{2}
29 1+(1.77−4.29i)T+(−20.5−20.5i)T2 1 + (1.77 - 4.29i)T + (-20.5 - 20.5i)T^{2}
31 1+(−6.72−2.78i)T+(21.9+21.9i)T2 1 + (-6.72 - 2.78i)T + (21.9 + 21.9i)T^{2}
37 1+(−8.58−3.55i)T+(26.1+26.1i)T2 1 + (-8.58 - 3.55i)T + (26.1 + 26.1i)T^{2}
41 1+(−0.0547−0.132i)T+(−28.9+28.9i)T2 1 + (-0.0547 - 0.132i)T + (-28.9 + 28.9i)T^{2}
43 1+(−0.587−0.587i)T+43iT2 1 + (-0.587 - 0.587i)T + 43iT^{2}
47 1+5.85iT−47T2 1 + 5.85iT - 47T^{2}
53 1+(6.54−6.54i)T−53iT2 1 + (6.54 - 6.54i)T - 53iT^{2}
59 1+(3.33+3.33i)T+59iT2 1 + (3.33 + 3.33i)T + 59iT^{2}
61 1+(−4.76−11.4i)T+(−43.1+43.1i)T2 1 + (-4.76 - 11.4i)T + (-43.1 + 43.1i)T^{2}
67 1+8.66T+67T2 1 + 8.66T + 67T^{2}
71 1+(−4.75−1.96i)T+(50.2+50.2i)T2 1 + (-4.75 - 1.96i)T + (50.2 + 50.2i)T^{2}
73 1+(6.17−14.9i)T+(−51.6−51.6i)T2 1 + (6.17 - 14.9i)T + (-51.6 - 51.6i)T^{2}
79 1+(−9.34+3.87i)T+(55.8−55.8i)T2 1 + (-9.34 + 3.87i)T + (55.8 - 55.8i)T^{2}
83 1+(2.40−2.40i)T−83iT2 1 + (2.40 - 2.40i)T - 83iT^{2}
89 1+2.24iT−89T2 1 + 2.24iT - 89T^{2}
97 1+(1.66−4.02i)T+(−68.5−68.5i)T2 1 + (1.66 - 4.02i)T + (-68.5 - 68.5i)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

−10.39136314373745539695713119847, −9.547920109272276048067065633042, −8.752994766607752852590070609224, −7.50241731393607983268213750826, −6.91922700656965797873005662451, −6.42282177425362199198629516088, −5.13333097879968832707519385089, −4.40930593169148692764128518556, −2.76846567329703591282255697690, −0.999144926813167731873336595678, 0.33230148423982085485867774265, 2.47058128529420839207777419673, 3.13822349389428564592013926372, 4.78019039079649257255423248709, 5.68697122384459393275189414103, 6.14746364039288646483713668571, 7.84495313551074021322823287595, 8.247232803089244709464875038218, 9.313037006437104917018778361587, 10.11967174917034704168827246090

Graph of the ZZ-function along the critical line