↑

Properties

Label 2-850-85.23-c1-0-19
Degree 22
Conductor 850850
Sign −0.281+0.959i-0.281 + 0.959i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 + 0.382i)2-s + (0.628 + 0.419i)3-s + (0.707 − 0.707i)4-s + (−0.741 − 0.147i)6-s + (−1.27 − 0.252i)7-s + (−0.382 + 0.923i)8-s + (−0.929 − 2.24i)9-s + (−0.546 + 2.74i)11-s + (0.741 − 0.147i)12-s − 4.10·13-s + (1.27 − 0.252i)14-s − i·16-s + (2.50 − 3.27i)17-s + (1.71 + 1.71i)18-s + (1.17 − 2.83i)19-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (0.362 + 0.242i)3-s + (0.353 − 0.353i)4-s + (−0.302 − 0.0601i)6-s + (−0.480 − 0.0955i)7-s + (−0.135 + 0.326i)8-s + (−0.309 − 0.747i)9-s + (−0.164 + 0.827i)11-s + (0.213 − 0.0425i)12-s − 1.13·13-s + (0.339 − 0.0675i)14-s − 0.250i·16-s + (0.607 − 0.794i)17-s + (0.404 + 0.404i)18-s + (0.269 − 0.650i)19-s + ⋯

Functional equation

Λ(s)=(850s/2 ΓC(s) L(s)=((−0.281+0.959i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(850s/2 ΓC(s+1/2) L(s)=((−0.281+0.959i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 850850    =    2⋅52⋅172 \cdot 5^{2} \cdot 17
Sign: −0.281+0.959i-0.281 + 0.959i
Analytic conductor: 6.787286.78728
Root analytic conductor: 2.605242.60524
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ850(193,⋅)\chi_{850} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 850, ( :1/2), −0.281+0.959i)(2,\ 850,\ (\ :1/2),\ -0.281 + 0.959i)

Particular Values

L(1)L(1) ≈\approx 0.319472−0.426693i0.319472 - 0.426693i
L(12)L(\frac12) ≈\approx 0.319472−0.426693i0.319472 - 0.426693i
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+(−2.50+3.27i)T 1 + (-2.50 + 3.27i)T
good3 1+(−0.628−0.419i)T+(1.14+2.77i)T2 1 + (-0.628 - 0.419i)T + (1.14 + 2.77i)T^{2}
7 1+(1.27+0.252i)T+(6.46+2.67i)T2 1 + (1.27 + 0.252i)T + (6.46 + 2.67i)T^{2}
11 1+(0.546−2.74i)T+(−10.1−4.20i)T2 1 + (0.546 - 2.74i)T + (-10.1 - 4.20i)T^{2}
13 1+4.10T+13T2 1 + 4.10T + 13T^{2}
19 1+(−1.17+2.83i)T+(−13.4−13.4i)T2 1 + (-1.17 + 2.83i)T + (-13.4 - 13.4i)T^{2}
23 1+(4.19+6.28i)T+(−8.80+21.2i)T2 1 + (4.19 + 6.28i)T + (-8.80 + 21.2i)T^{2}
29 1+(3.51−5.26i)T+(−11.0−26.7i)T2 1 + (3.51 - 5.26i)T + (-11.0 - 26.7i)T^{2}
31 1+(0.355+1.78i)T+(−28.6+11.8i)T2 1 + (0.355 + 1.78i)T + (-28.6 + 11.8i)T^{2}
37 1+(0.190−0.284i)T+(−14.1−34.1i)T2 1 + (0.190 - 0.284i)T + (-14.1 - 34.1i)T^{2}
41 1+(1.73+2.59i)T+(−15.6+37.8i)T2 1 + (1.73 + 2.59i)T + (-15.6 + 37.8i)T^{2}
43 1+(7.35+3.04i)T+(30.4+30.4i)T2 1 + (7.35 + 3.04i)T + (30.4 + 30.4i)T^{2}
47 1+7.90iT−47T2 1 + 7.90iT - 47T^{2}
53 1+(−4.29−10.3i)T+(−37.4+37.4i)T2 1 + (-4.29 - 10.3i)T + (-37.4 + 37.4i)T^{2}
59 1+(−9.44+3.91i)T+(41.7−41.7i)T2 1 + (-9.44 + 3.91i)T + (41.7 - 41.7i)T^{2}
61 1+(−5.15+3.44i)T+(23.3−56.3i)T2 1 + (-5.15 + 3.44i)T + (23.3 - 56.3i)T^{2}
67 1+(0.944+0.944i)T+67iT2 1 + (0.944 + 0.944i)T + 67iT^{2}
71 1+(7.94−1.58i)T+(65.5−27.1i)T2 1 + (7.94 - 1.58i)T + (65.5 - 27.1i)T^{2}
73 1+(1.75−0.349i)T+(67.4−27.9i)T2 1 + (1.75 - 0.349i)T + (67.4 - 27.9i)T^{2}
79 1+(16.2+3.23i)T+(72.9+30.2i)T2 1 + (16.2 + 3.23i)T + (72.9 + 30.2i)T^{2}
83 1+(−3.31+1.37i)T+(58.6−58.6i)T2 1 + (-3.31 + 1.37i)T + (58.6 - 58.6i)T^{2}
89 1+(−7.01+7.01i)T−89iT2 1 + (-7.01 + 7.01i)T - 89iT^{2}
97 1+(8.19−1.63i)T+(89.6−37.1i)T2 1 + (8.19 - 1.63i)T + (89.6 - 37.1i)T^{2}
show more
show less
   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

−9.893253682732586701497368943587, −9.171688406773251078255812981173, −8.386675501295990192293901100676, −7.25975263092855914486360377308, −6.83461752961745385717865530140, −5.60349489244370163774014540937, −4.59914267834896282871534134212, −3.27456221313210193703170769686, −2.24578239173049638906999663725, −0.28871482091835882311999453139, 1.68429561528171041505709818085, 2.81105566894806176927570429018, 3.73892693970953727838659584573, 5.30936889095665157263751183274, 6.12290161823223649146936682143, 7.39783660627566440304490188810, 7.952612767480331888778209347813, 8.644956482135533923505753724546, 9.825077510270002715582173028513, 10.08672796266128276833229751383

Graph of the ZZ-function along the critical line