Properties

Label 2-616-77.41-c1-0-21
Degree 22
Conductor 616616
Sign 0.646+0.763i0.646 + 0.763i
Analytic cond. 4.918784.91878
Root an. cond. 2.217832.21783
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  + (1.74 − 0.565i)3-s + (2.00 − 2.75i)5-s + (2.13 − 1.56i)7-s + (0.282 − 0.205i)9-s + (1.95 + 2.67i)11-s + (−3.12 + 2.26i)13-s + (1.92 − 5.92i)15-s + (5.18 + 3.76i)17-s + (−1.35 − 4.18i)19-s + (2.83 − 3.92i)21-s − 8.95·23-s + (−2.03 − 6.27i)25-s + (−2.85 + 3.92i)27-s + (−2.95 − 0.961i)29-s + (−0.176 − 0.242i)31-s + ⋯
L(s)  = 1  + (1.00 − 0.326i)3-s + (0.895 − 1.23i)5-s + (0.807 − 0.590i)7-s + (0.0942 − 0.0684i)9-s + (0.589 + 0.807i)11-s + (−0.865 + 0.629i)13-s + (0.497 − 1.53i)15-s + (1.25 + 0.913i)17-s + (−0.311 − 0.959i)19-s + (0.618 − 0.856i)21-s − 1.86·23-s + (−0.407 − 1.25i)25-s + (−0.548 + 0.755i)27-s + (−0.549 − 0.178i)29-s + (−0.0316 − 0.0436i)31-s + ⋯

Functional equation

Λ(s)=(616s/2ΓC(s)L(s)=((0.646+0.763i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(616s/2ΓC(s+1/2)L(s)=((0.646+0.763i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 616616    =    237112^{3} \cdot 7 \cdot 11
Sign: 0.646+0.763i0.646 + 0.763i
Analytic conductor: 4.918784.91878
Root analytic conductor: 2.217832.21783
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ616(41,)\chi_{616} (41, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 616, ( :1/2), 0.646+0.763i)(2,\ 616,\ (\ :1/2),\ 0.646 + 0.763i)

Particular Values

L(1)L(1) \approx 2.220951.02992i2.22095 - 1.02992i
L(12)L(\frac12) \approx 2.220951.02992i2.22095 - 1.02992i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1+(2.13+1.56i)T 1 + (-2.13 + 1.56i)T
11 1+(1.952.67i)T 1 + (-1.95 - 2.67i)T
good3 1+(1.74+0.565i)T+(2.421.76i)T2 1 + (-1.74 + 0.565i)T + (2.42 - 1.76i)T^{2}
5 1+(2.00+2.75i)T+(1.544.75i)T2 1 + (-2.00 + 2.75i)T + (-1.54 - 4.75i)T^{2}
13 1+(3.122.26i)T+(4.0112.3i)T2 1 + (3.12 - 2.26i)T + (4.01 - 12.3i)T^{2}
17 1+(5.183.76i)T+(5.25+16.1i)T2 1 + (-5.18 - 3.76i)T + (5.25 + 16.1i)T^{2}
19 1+(1.35+4.18i)T+(15.3+11.1i)T2 1 + (1.35 + 4.18i)T + (-15.3 + 11.1i)T^{2}
23 1+8.95T+23T2 1 + 8.95T + 23T^{2}
29 1+(2.95+0.961i)T+(23.4+17.0i)T2 1 + (2.95 + 0.961i)T + (23.4 + 17.0i)T^{2}
31 1+(0.176+0.242i)T+(9.57+29.4i)T2 1 + (0.176 + 0.242i)T + (-9.57 + 29.4i)T^{2}
37 1+(0.331+1.02i)T+(29.921.7i)T2 1 + (-0.331 + 1.02i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.1310.404i)T+(33.1+24.0i)T2 1 + (-0.131 - 0.404i)T + (-33.1 + 24.0i)T^{2}
43 19.11iT43T2 1 - 9.11iT - 43T^{2}
47 1+(2.540.827i)T+(38.027.6i)T2 1 + (2.54 - 0.827i)T + (38.0 - 27.6i)T^{2}
53 1+(10.0+7.30i)T+(16.350.4i)T2 1 + (-10.0 + 7.30i)T + (16.3 - 50.4i)T^{2}
59 1+(3.66+1.18i)T+(47.7+34.6i)T2 1 + (3.66 + 1.18i)T + (47.7 + 34.6i)T^{2}
61 1+(8.456.14i)T+(18.8+58.0i)T2 1 + (-8.45 - 6.14i)T + (18.8 + 58.0i)T^{2}
67 1+1.32T+67T2 1 + 1.32T + 67T^{2}
71 1+(12.9+9.37i)T+(21.9+67.5i)T2 1 + (12.9 + 9.37i)T + (21.9 + 67.5i)T^{2}
73 1+(2.206.77i)T+(59.042.9i)T2 1 + (2.20 - 6.77i)T + (-59.0 - 42.9i)T^{2}
79 1+(4.315.94i)T+(24.4+75.1i)T2 1 + (-4.31 - 5.94i)T + (-24.4 + 75.1i)T^{2}
83 1+(0.928+0.674i)T+(25.6+78.9i)T2 1 + (0.928 + 0.674i)T + (25.6 + 78.9i)T^{2}
89 1+3.58iT89T2 1 + 3.58iT - 89T^{2}
97 1+(1.20+1.65i)T+(29.9+92.2i)T2 1 + (1.20 + 1.65i)T + (-29.9 + 92.2i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.10628694817537808631023462740, −9.588151129042187082760964612581, −8.706613222226726535144721591924, −8.011948849369435795837251861778, −7.22056962723936811426904177886, −5.87416383749755014459238773427, −4.82914366940291664991842104433, −3.99527891931610275652458765780, −2.18345622779691534716984027969, −1.51819912626969854230709132097, 2.03345409597298562631132211936, 2.88916095704423994789125952446, 3.79575808576348509777930088690, 5.49853462174348065292433864992, 6.04732942122309693990985521909, 7.42299266329557992300039935456, 8.140601228039087312642584442179, 9.068118442068037612224528390108, 9.944404554082379375194516988320, 10.40403839362428562366955761120

Graph of the ZZ-function along the critical line