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Properties

Label 2-29e2-29.5-c1-0-5
Degree 22
Conductor 841841
Sign −0.565−0.824i-0.565 - 0.824i
Analytic cond. 6.715416.71541
Root an. cond. 2.591412.59141
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  + (0.483 − 0.385i)2-s + (0.602 − 0.137i)3-s + (−0.360 + 1.57i)4-s + (−2.40 − 3.01i)5-s + (0.238 − 0.298i)6-s + (0.497 + 2.18i)7-s + (0.970 + 2.01i)8-s + (−2.35 + 1.13i)9-s + (−2.32 − 0.530i)10-s + (0.599 − 1.24i)11-s + i·12-s + (−0.212 − 0.102i)13-s + (1.08 + 0.861i)14-s + (−1.86 − 1.48i)15-s + (−1.67 − 0.804i)16-s + 4.38i·17-s + ⋯
L(s)  = 1  + (0.341 − 0.272i)2-s + (0.347 − 0.0794i)3-s + (−0.180 + 0.788i)4-s + (−1.07 − 1.34i)5-s + (0.0972 − 0.121i)6-s + (0.188 + 0.823i)7-s + (0.343 + 0.712i)8-s + (−0.786 + 0.378i)9-s + (−0.734 − 0.167i)10-s + (0.180 − 0.375i)11-s + 0.288i·12-s + (−0.0589 − 0.0284i)13-s + (0.288 + 0.230i)14-s + (−0.480 − 0.383i)15-s + (−0.417 − 0.201i)16-s + 1.06i·17-s + ⋯

Functional equation

Λ(s)=(841s/2 ΓC(s) L(s)=((−0.565−0.824i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(841s/2 ΓC(s+1/2) L(s)=((−0.565−0.824i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 841841    =    29229^{2}
Sign: −0.565−0.824i-0.565 - 0.824i
Analytic conductor: 6.715416.71541
Root analytic conductor: 2.591412.59141
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ841(63,⋅)\chi_{841} (63, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 841, ( :1/2), −0.565−0.824i)(2,\ 841,\ (\ :1/2),\ -0.565 - 0.824i)

Particular Values

L(1)L(1) ≈\approx 0.306850+0.582156i0.306850 + 0.582156i
L(12)L(\frac12) ≈\approx 0.306850+0.582156i0.306850 + 0.582156i
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)
bad29 1 1
good2 1+(−0.483+0.385i)T+(0.445−1.94i)T2 1 + (-0.483 + 0.385i)T + (0.445 - 1.94i)T^{2}
3 1+(−0.602+0.137i)T+(2.70−1.30i)T2 1 + (-0.602 + 0.137i)T + (2.70 - 1.30i)T^{2}
5 1+(2.40+3.01i)T+(−1.11+4.87i)T2 1 + (2.40 + 3.01i)T + (-1.11 + 4.87i)T^{2}
7 1+(−0.497−2.18i)T+(−6.30+3.03i)T2 1 + (-0.497 - 2.18i)T + (-6.30 + 3.03i)T^{2}
11 1+(−0.599+1.24i)T+(−6.85−8.60i)T2 1 + (-0.599 + 1.24i)T + (-6.85 - 8.60i)T^{2}
13 1+(0.212+0.102i)T+(8.10+10.1i)T2 1 + (0.212 + 0.102i)T + (8.10 + 10.1i)T^{2}
17 1−4.38iT−17T2 1 - 4.38iT - 17T^{2}
19 1+(4.73+1.08i)T+(17.1+8.24i)T2 1 + (4.73 + 1.08i)T + (17.1 + 8.24i)T^{2}
23 1+(0.770−0.966i)T+(−5.11−22.4i)T2 1 + (0.770 - 0.966i)T + (-5.11 - 22.4i)T^{2}
31 1+(7.88−6.29i)T+(6.89−30.2i)T2 1 + (7.88 - 6.29i)T + (6.89 - 30.2i)T^{2}
37 1+(2.04+4.24i)T+(−23.0+28.9i)T2 1 + (2.04 + 4.24i)T + (-23.0 + 28.9i)T^{2}
41 1−3.85iT−41T2 1 - 3.85iT - 41T^{2}
43 1+(−5.65−4.51i)T+(9.56+41.9i)T2 1 + (-5.65 - 4.51i)T + (9.56 + 41.9i)T^{2}
47 1+(3.03−6.30i)T+(−29.3−36.7i)T2 1 + (3.03 - 6.30i)T + (-29.3 - 36.7i)T^{2}
53 1+(1.24+1.56i)T+(−11.7+51.6i)T2 1 + (1.24 + 1.56i)T + (-11.7 + 51.6i)T^{2}
59 1−6.09T+59T2 1 - 6.09T + 59T^{2}
61 1+(−0.602+0.137i)T+(54.9−26.4i)T2 1 + (-0.602 + 0.137i)T + (54.9 - 26.4i)T^{2}
67 1+(1.37−0.662i)T+(41.7−52.3i)T2 1 + (1.37 - 0.662i)T + (41.7 - 52.3i)T^{2}
71 1+(−9.43−4.54i)T+(44.2+55.5i)T2 1 + (-9.43 - 4.54i)T + (44.2 + 55.5i)T^{2}
73 1+(10.7+8.54i)T+(16.2+71.1i)T2 1 + (10.7 + 8.54i)T + (16.2 + 71.1i)T^{2}
79 1+(2.64+5.48i)T+(−49.2+61.7i)T2 1 + (2.64 + 5.48i)T + (-49.2 + 61.7i)T^{2}
83 1+(−2.21+9.69i)T+(−74.7−36.0i)T2 1 + (-2.21 + 9.69i)T + (-74.7 - 36.0i)T^{2}
89 1+(−3.68+2.93i)T+(19.8−86.7i)T2 1 + (-3.68 + 2.93i)T + (19.8 - 86.7i)T^{2}
97 1+(3.47+0.792i)T+(87.3+42.0i)T2 1 + (3.47 + 0.792i)T + (87.3 + 42.0i)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

−10.90510885836201167127089234412, −9.111267277900372621937855259255, −8.645078161733281963902768104852, −8.252756480572769723726063788019, −7.45783737909508928540323009919, −5.84652687668380967870102493105, −4.95663990893561782618482424371, −4.10515849839985216788114537371, −3.23249779465590422312468948117, −1.93300539523557258505901627711, 0.26276323696761564905778988739, 2.38899737439150395543751648070, 3.70529036693392654659806046813, 4.21622049898435331068550929819, 5.54210754087037635665342341305, 6.65590921620961194637362262718, 7.13200685429224085106084946167, 8.007961975748728325412589834558, 9.118905893325525693517710241744, 10.05313508521739053352009388103

Graph of the ZZ-function along the critical line