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Properties

Label 2-56-56.53-c3-0-14
Degree 22
Conductor 5656
Sign 0.936+0.350i0.936 + 0.350i
Analytic cond. 3.304103.30410
Root an. cond. 1.817721.81772
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.53 + 1.25i)2-s + (−7.07 − 4.08i)3-s + (4.85 + 6.35i)4-s + (17.0 − 9.82i)5-s + (−12.8 − 19.2i)6-s + (15.6 − 9.83i)7-s + (4.34 + 22.2i)8-s + (19.9 + 34.4i)9-s + (55.4 − 3.57i)10-s + (−29.7 − 17.1i)11-s + (−8.39 − 64.8i)12-s − 3.09i·13-s + (52.1 − 5.26i)14-s − 160.·15-s + (−16.8 + 61.7i)16-s + (−14.8 + 25.6i)17-s + ⋯
L(s)  = 1  + (0.896 + 0.443i)2-s + (−1.36 − 0.786i)3-s + (0.607 + 0.794i)4-s + (1.52 − 0.879i)5-s + (−0.872 − 1.30i)6-s + (0.847 − 0.531i)7-s + (0.191 + 0.981i)8-s + (0.737 + 1.27i)9-s + (1.75 − 0.113i)10-s + (−0.814 − 0.470i)11-s + (−0.202 − 1.56i)12-s − 0.0661i·13-s + (0.994 − 0.100i)14-s − 2.76·15-s + (−0.262 + 0.964i)16-s + (−0.211 + 0.365i)17-s + ⋯

Functional equation

Λ(s)=(56s/2 ΓC(s) L(s)=((0.936+0.350i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(56s/2 ΓC(s+3/2) L(s)=((0.936+0.350i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5656    =    23⋅72^{3} \cdot 7
Sign: 0.936+0.350i0.936 + 0.350i
Analytic conductor: 3.304103.30410
Root analytic conductor: 1.817721.81772
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ56(53,⋅)\chi_{56} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 56, ( :3/2), 0.936+0.350i)(2,\ 56,\ (\ :3/2),\ 0.936 + 0.350i)

Particular Values

L(2)L(2) ≈\approx 1.88937−0.341468i1.88937 - 0.341468i
L(12)L(\frac12) ≈\approx 1.88937−0.341468i1.88937 - 0.341468i
L(52)L(\frac{5}{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+(−2.53−1.25i)T 1 + (-2.53 - 1.25i)T
7 1+(−15.6+9.83i)T 1 + (-15.6 + 9.83i)T
good3 1+(7.07+4.08i)T+(13.5+23.3i)T2 1 + (7.07 + 4.08i)T + (13.5 + 23.3i)T^{2}
5 1+(−17.0+9.82i)T+(62.5−108.i)T2 1 + (-17.0 + 9.82i)T + (62.5 - 108. i)T^{2}
11 1+(29.7+17.1i)T+(665.5+1.15e3i)T2 1 + (29.7 + 17.1i)T + (665.5 + 1.15e3i)T^{2}
13 1+3.09iT−2.19e3T2 1 + 3.09iT - 2.19e3T^{2}
17 1+(14.8−25.6i)T+(−2.45e3−4.25e3i)T2 1 + (14.8 - 25.6i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(40.0−23.1i)T+(3.42e3−5.94e3i)T2 1 + (40.0 - 23.1i)T + (3.42e3 - 5.94e3i)T^{2}
23 1+(−49.7−86.0i)T+(−6.08e3+1.05e4i)T2 1 + (-49.7 - 86.0i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1−136.iT−2.43e4T2 1 - 136. iT - 2.43e4T^{2}
31 1+(0.290−0.502i)T+(−1.48e4−2.57e4i)T2 1 + (0.290 - 0.502i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(128.−74.0i)T+(2.53e4−4.38e4i)T2 1 + (128. - 74.0i)T + (2.53e4 - 4.38e4i)T^{2}
41 1+141.T+6.89e4T2 1 + 141.T + 6.89e4T^{2}
43 1−259.iT−7.95e4T2 1 - 259. iT - 7.95e4T^{2}
47 1+(174.+301.i)T+(−5.19e4+8.99e4i)T2 1 + (174. + 301. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(184.+106.i)T+(7.44e4+1.28e5i)T2 1 + (184. + 106. i)T + (7.44e4 + 1.28e5i)T^{2}
59 1+(100.+57.9i)T+(1.02e5+1.77e5i)T2 1 + (100. + 57.9i)T + (1.02e5 + 1.77e5i)T^{2}
61 1+(−345.+199.i)T+(1.13e5−1.96e5i)T2 1 + (-345. + 199. i)T + (1.13e5 - 1.96e5i)T^{2}
67 1+(−893.−515.i)T+(1.50e5+2.60e5i)T2 1 + (-893. - 515. i)T + (1.50e5 + 2.60e5i)T^{2}
71 1−476.T+3.57e5T2 1 - 476.T + 3.57e5T^{2}
73 1+(554.−960.i)T+(−1.94e5−3.36e5i)T2 1 + (554. - 960. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(225.+391.i)T+(−2.46e5+4.26e5i)T2 1 + (225. + 391. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+770.iT−5.71e5T2 1 + 770. iT - 5.71e5T^{2}
89 1+(615.+1.06e3i)T+(−3.52e5+6.10e5i)T2 1 + (615. + 1.06e3i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−706.T+9.12e5T2 1 - 706.T + 9.12e5T^{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

−14.33823025226148858139288701689, −13.22992477061628049643505786560, −12.86473471696076590446911418042, −11.52296595998513668940167547032, −10.47773950860523813929262868232, −8.311057036604191730516753763408, −6.78949071914508261647494819264, −5.61320224826642492280642668386, −5.00425526259837835091751741928, −1.62597726957999671798427521303, 2.31725461991407600992767223249, 4.80466372186571456559953868753, 5.61650252154704716553628556899, 6.65595958917547814699092960873, 9.650635631792153709694176667084, 10.58746922861339757268433832820, 11.15319685403056850462826234191, 12.43397460743115742269124528282, 13.71560686418139426657630261058, 14.81877921670701251769938842682

Graph of the ZZ-function along the critical line