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Properties

Label 2-850-85.37-c1-0-22
Degree 22
Conductor 850850
Sign 0.156+0.987i0.156 + 0.987i
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  + (−0.923 − 0.382i)2-s + (1.98 − 1.32i)3-s + (0.707 + 0.707i)4-s + (−2.34 + 0.465i)6-s + (4.88 − 0.972i)7-s + (−0.382 − 0.923i)8-s + (1.03 − 2.49i)9-s + (−0.514 − 2.58i)11-s + (2.34 + 0.465i)12-s − 3.55·13-s + (−4.88 − 0.972i)14-s + i·16-s + (3.56 − 2.06i)17-s + (−1.91 + 1.91i)18-s + (−1.27 − 3.08i)19-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (1.14 − 0.766i)3-s + (0.353 + 0.353i)4-s + (−0.956 + 0.190i)6-s + (1.84 − 0.367i)7-s + (−0.135 − 0.326i)8-s + (0.345 − 0.832i)9-s + (−0.155 − 0.779i)11-s + (0.676 + 0.134i)12-s − 0.986·13-s + (−1.30 − 0.259i)14-s + 0.250i·16-s + (0.865 − 0.500i)17-s + (−0.450 + 0.450i)18-s + (−0.293 − 0.707i)19-s + ⋯

Functional equation

Λ(s)=(850s/2 ΓC(s) L(s)=((0.156+0.987i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(850s/2 ΓC(s+1/2) L(s)=((0.156+0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 850850    =    2⋅52⋅172 \cdot 5^{2} \cdot 17
Sign: 0.156+0.987i0.156 + 0.987i
Analytic conductor: 6.787286.78728
Root analytic conductor: 2.605242.60524
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ850(207,⋅)\chi_{850} (207, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 850, ( :1/2), 0.156+0.987i)(2,\ 850,\ (\ :1/2),\ 0.156 + 0.987i)

Particular Values

L(1)L(1) ≈\approx 1.48774−1.27046i1.48774 - 1.27046i
L(12)L(\frac12) ≈\approx 1.48774−1.27046i1.48774 - 1.27046i
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+(0.923+0.382i)T 1 + (0.923 + 0.382i)T
5 1 1
17 1+(−3.56+2.06i)T 1 + (-3.56 + 2.06i)T
good3 1+(−1.98+1.32i)T+(1.14−2.77i)T2 1 + (-1.98 + 1.32i)T + (1.14 - 2.77i)T^{2}
7 1+(−4.88+0.972i)T+(6.46−2.67i)T2 1 + (-4.88 + 0.972i)T + (6.46 - 2.67i)T^{2}
11 1+(0.514+2.58i)T+(−10.1+4.20i)T2 1 + (0.514 + 2.58i)T + (-10.1 + 4.20i)T^{2}
13 1+3.55T+13T2 1 + 3.55T + 13T^{2}
19 1+(1.27+3.08i)T+(−13.4+13.4i)T2 1 + (1.27 + 3.08i)T + (-13.4 + 13.4i)T^{2}
23 1+(3.96−5.93i)T+(−8.80−21.2i)T2 1 + (3.96 - 5.93i)T + (-8.80 - 21.2i)T^{2}
29 1+(−0.593−0.888i)T+(−11.0+26.7i)T2 1 + (-0.593 - 0.888i)T + (-11.0 + 26.7i)T^{2}
31 1+(−0.489+2.46i)T+(−28.6−11.8i)T2 1 + (-0.489 + 2.46i)T + (-28.6 - 11.8i)T^{2}
37 1+(−4.15−6.21i)T+(−14.1+34.1i)T2 1 + (-4.15 - 6.21i)T + (-14.1 + 34.1i)T^{2}
41 1+(1.27−1.90i)T+(−15.6−37.8i)T2 1 + (1.27 - 1.90i)T + (-15.6 - 37.8i)T^{2}
43 1+(−7.30+3.02i)T+(30.4−30.4i)T2 1 + (-7.30 + 3.02i)T + (30.4 - 30.4i)T^{2}
47 1+7.15iT−47T2 1 + 7.15iT - 47T^{2}
53 1+(2.62−6.34i)T+(−37.4−37.4i)T2 1 + (2.62 - 6.34i)T + (-37.4 - 37.4i)T^{2}
59 1+(9.79+4.05i)T+(41.7+41.7i)T2 1 + (9.79 + 4.05i)T + (41.7 + 41.7i)T^{2}
61 1+(−5.08−3.39i)T+(23.3+56.3i)T2 1 + (-5.08 - 3.39i)T + (23.3 + 56.3i)T^{2}
67 1+(8.80−8.80i)T−67iT2 1 + (8.80 - 8.80i)T - 67iT^{2}
71 1+(14.0+2.80i)T+(65.5+27.1i)T2 1 + (14.0 + 2.80i)T + (65.5 + 27.1i)T^{2}
73 1+(−7.24−1.44i)T+(67.4+27.9i)T2 1 + (-7.24 - 1.44i)T + (67.4 + 27.9i)T^{2}
79 1+(5.45−1.08i)T+(72.9−30.2i)T2 1 + (5.45 - 1.08i)T + (72.9 - 30.2i)T^{2}
83 1+(−3.02−1.25i)T+(58.6+58.6i)T2 1 + (-3.02 - 1.25i)T + (58.6 + 58.6i)T^{2}
89 1+(4.91+4.91i)T+89iT2 1 + (4.91 + 4.91i)T + 89iT^{2}
97 1+(−13.4−2.67i)T+(89.6+37.1i)T2 1 + (-13.4 - 2.67i)T + (89.6 + 37.1i)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

−9.855309289604606527296505959962, −8.928000714967611564969027247025, −8.191715259428608411689385632193, −7.67382136761168430330081317396, −7.19496171000096475586070817956, −5.60084658438281316664497645488, −4.46673926187059547533094104407, −3.09462401186672588948920841409, −2.14017662505684706364471675671, −1.15691005305367420519962964803, 1.77783014457140798390379649022, 2.60320206593345019673141227393, 4.20324097219625776917288777020, 4.84344157594028766636887834073, 5.98780917953767037901170982773, 7.59408308928814992570692069078, 7.903715837686957013441616418322, 8.632364319863303315150134455965, 9.424893999222770160620409381284, 10.21833813789103204942483475117

Graph of the ZZ-function along the critical line