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Properties

Label 2-29e2-29.5-c1-0-21
Degree 22
Conductor 841841
Sign 0.959+0.280i0.959 + 0.280i
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)=((0.959+0.280i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(841s/2 ΓC(s+1/2) L(s)=((0.959+0.280i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 841841    =    29229^{2}
Sign: 0.959+0.280i0.959 + 0.280i
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), 0.959+0.280i)(2,\ 841,\ (\ :1/2),\ 0.959 + 0.280i)

Particular Values

L(1)L(1) ≈\approx 0.622336−0.0891486i0.622336 - 0.0891486i
L(12)L(\frac12) ≈\approx 0.622336−0.0891486i0.622336 - 0.0891486i
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

−9.832348261910190735039300449071, −9.100904833836167295903142168594, −8.275370589070684939729164820063, −7.990778815317527424748306812936, −6.97684769378808574787813411443, −5.48313722327801329065921412930, −4.92794029567890331250617640142, −3.90113115861377640154787546920, −2.65719264078806938186013227484, −0.51566302638163570410710263818, 0.889337828841345916381451073173, 2.75928883465164406320496346467, 3.68077664947873830120129260825, 4.87403521096344484593562204629, 6.10111557418630117440157518870, 6.68869530819481486162210016170, 7.75511910666771003929141104723, 8.443171301947526139891541334559, 9.631543538772115195636821705381, 10.57377543277009793512584360486

Graph of the ZZ-function along the critical line