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Properties

Label 2-859-859.100-c1-0-28
Degree 22
Conductor 859859
Sign 0.311−0.950i0.311 - 0.950i
Analytic cond. 6.859146.85914
Root an. cond. 2.618992.61899
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  + (−1.37 + 0.719i)2-s + (−0.554 + 0.803i)3-s + (0.223 − 0.324i)4-s + (0.478 + 0.692i)5-s + (0.181 − 1.49i)6-s + (0.117 + 0.969i)7-s + (0.299 − 2.46i)8-s + (0.726 + 1.91i)9-s + (−1.15 − 0.605i)10-s + (−0.555 − 4.57i)11-s + (0.136 + 0.359i)12-s + 2.58·13-s + (−0.858 − 1.24i)14-s − 0.821·15-s + (1.64 + 4.33i)16-s + (7.12 − 1.75i)17-s + ⋯
L(s)  = 1  + (−0.968 + 0.508i)2-s + (−0.320 + 0.463i)3-s + (0.111 − 0.162i)4-s + (0.213 + 0.309i)5-s + (0.0742 − 0.611i)6-s + (0.0444 + 0.366i)7-s + (0.105 − 0.872i)8-s + (0.242 + 0.638i)9-s + (−0.364 − 0.191i)10-s + (−0.167 − 1.37i)11-s + (0.0393 + 0.103i)12-s + 0.717·13-s + (−0.229 − 0.332i)14-s − 0.212·15-s + (0.410 + 1.08i)16-s + (1.72 − 0.425i)17-s + ⋯

Functional equation

Λ(s)=(859s/2 ΓC(s) L(s)=((0.311−0.950i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(859s/2 ΓC(s+1/2) L(s)=((0.311−0.950i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 859859
Sign: 0.311−0.950i0.311 - 0.950i
Analytic conductor: 6.859146.85914
Root analytic conductor: 2.618992.61899
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ859(100,⋅)\chi_{859} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 859, ( :1/2), 0.311−0.950i)(2,\ 859,\ (\ :1/2),\ 0.311 - 0.950i)

Particular Values

L(1)L(1) ≈\approx 0.732048+0.530512i0.732048 + 0.530512i
L(12)L(\frac12) ≈\approx 0.732048+0.530512i0.732048 + 0.530512i
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)
bad859 1+(−22.1−19.2i)T 1 + (-22.1 - 19.2i)T
good2 1+(1.37−0.719i)T+(1.13−1.64i)T2 1 + (1.37 - 0.719i)T + (1.13 - 1.64i)T^{2}
3 1+(0.554−0.803i)T+(−1.06−2.80i)T2 1 + (0.554 - 0.803i)T + (-1.06 - 2.80i)T^{2}
5 1+(−0.478−0.692i)T+(−1.77+4.67i)T2 1 + (-0.478 - 0.692i)T + (-1.77 + 4.67i)T^{2}
7 1+(−0.117−0.969i)T+(−6.79+1.67i)T2 1 + (-0.117 - 0.969i)T + (-6.79 + 1.67i)T^{2}
11 1+(0.555+4.57i)T+(−10.6+2.63i)T2 1 + (0.555 + 4.57i)T + (-10.6 + 2.63i)T^{2}
13 1−2.58T+13T2 1 - 2.58T + 13T^{2}
17 1+(−7.12+1.75i)T+(15.0−7.90i)T2 1 + (-7.12 + 1.75i)T + (15.0 - 7.90i)T^{2}
19 1+0.113T+19T2 1 + 0.113T + 19T^{2}
23 1+(−1.35−0.709i)T+(13.0+18.9i)T2 1 + (-1.35 - 0.709i)T + (13.0 + 18.9i)T^{2}
29 1+(−2.43+6.41i)T+(−21.7−19.2i)T2 1 + (-2.43 + 6.41i)T + (-21.7 - 19.2i)T^{2}
31 1+(−6.69−5.93i)T+(3.73+30.7i)T2 1 + (-6.69 - 5.93i)T + (3.73 + 30.7i)T^{2}
37 1+(2.90+7.64i)T+(−27.6+24.5i)T2 1 + (2.90 + 7.64i)T + (-27.6 + 24.5i)T^{2}
41 1+(0.748−1.08i)T+(−14.5−38.3i)T2 1 + (0.748 - 1.08i)T + (-14.5 - 38.3i)T^{2}
43 1+3.79T+43T2 1 + 3.79T + 43T^{2}
47 1+(5.64−5.00i)T+(5.66−46.6i)T2 1 + (5.64 - 5.00i)T + (5.66 - 46.6i)T^{2}
53 1+(−8.70−2.14i)T+(46.9+24.6i)T2 1 + (-8.70 - 2.14i)T + (46.9 + 24.6i)T^{2}
59 1+(−3.67+1.92i)T+(33.5−48.5i)T2 1 + (-3.67 + 1.92i)T + (33.5 - 48.5i)T^{2}
61 1+7.94T+61T2 1 + 7.94T + 61T^{2}
67 1+(−0.121−0.320i)T+(−50.1+44.4i)T2 1 + (-0.121 - 0.320i)T + (-50.1 + 44.4i)T^{2}
71 1+(−12.1−2.98i)T+(62.8+32.9i)T2 1 + (-12.1 - 2.98i)T + (62.8 + 32.9i)T^{2}
73 1+(1.35−11.1i)T+(−70.8−17.4i)T2 1 + (1.35 - 11.1i)T + (-70.8 - 17.4i)T^{2}
79 1+(3.89−5.63i)T+(−28.0−73.8i)T2 1 + (3.89 - 5.63i)T + (-28.0 - 73.8i)T^{2}
83 1+(−1.59−13.1i)T+(−80.5+19.8i)T2 1 + (-1.59 - 13.1i)T + (-80.5 + 19.8i)T^{2}
89 1+(5.66+1.39i)T+(78.8+41.3i)T2 1 + (5.66 + 1.39i)T + (78.8 + 41.3i)T^{2}
97 1+(9.81−2.41i)T+(85.8−45.0i)T2 1 + (9.81 - 2.41i)T + (85.8 - 45.0i)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

−10.20648082030122277013789406726, −9.547617975665443666959188560175, −8.390726567647743494759294978127, −8.165626726007702628712741199667, −7.03323395550288789295768303912, −6.04795639494841174694397677056, −5.31871583750931287454140306598, −3.97086455495975565485561509869, −2.89206635426169399097394116996, −0.950772013123686558563321334237, 0.988307189029650924524770206425, 1.70638433940474295705976126319, 3.33348652626468980786280989800, 4.72144524461135206433666750470, 5.64150002746572169873123010370, 6.75192261689519265804101820039, 7.59096865061430030013219150246, 8.437038786902136723517902288006, 9.338533644357968036090160856466, 10.06001126444314423468106498507

Graph of the ZZ-function along the critical line