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Properties

Label 2-56-56.37-c3-0-6
Degree 22
Conductor 5656
Sign 0.104+0.994i0.104 + 0.994i
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  + (−2.70 − 0.819i)2-s + (−8.54 + 4.93i)3-s + (6.65 + 4.43i)4-s + (3.19 + 1.84i)5-s + (27.1 − 6.35i)6-s + (−18.3 − 2.28i)7-s + (−14.3 − 17.4i)8-s + (35.2 − 60.9i)9-s + (−7.13 − 7.61i)10-s + (28.5 − 16.4i)11-s + (−78.8 − 5.07i)12-s − 33.0i·13-s + (47.8 + 21.2i)14-s − 36.4·15-s + (24.6 + 59.0i)16-s + (26.6 + 46.2i)17-s + ⋯
L(s)  = 1  + (−0.957 − 0.289i)2-s + (−1.64 + 0.949i)3-s + (0.832 + 0.554i)4-s + (0.285 + 0.165i)5-s + (1.84 − 0.432i)6-s + (−0.992 − 0.123i)7-s + (−0.635 − 0.771i)8-s + (1.30 − 2.25i)9-s + (−0.225 − 0.240i)10-s + (0.782 − 0.451i)11-s + (−1.89 − 0.122i)12-s − 0.704i·13-s + (0.913 + 0.405i)14-s − 0.626·15-s + (0.384 + 0.922i)16-s + (0.380 + 0.659i)17-s + ⋯

Functional equation

Λ(s)=(56s/2 ΓC(s) L(s)=((0.104+0.994i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(56s/2 ΓC(s+3/2) L(s)=((0.104+0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5656    =    23⋅72^{3} \cdot 7
Sign: 0.104+0.994i0.104 + 0.994i
Analytic conductor: 3.304103.30410
Root analytic conductor: 1.817721.81772
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ56(37,⋅)\chi_{56} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 56, ( :3/2), 0.104+0.994i)(2,\ 56,\ (\ :3/2),\ 0.104 + 0.994i)

Particular Values

L(2)L(2) ≈\approx 0.225427−0.202932i0.225427 - 0.202932i
L(12)L(\frac12) ≈\approx 0.225427−0.202932i0.225427 - 0.202932i
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+(2.70+0.819i)T 1 + (2.70 + 0.819i)T
7 1+(18.3+2.28i)T 1 + (18.3 + 2.28i)T
good3 1+(8.54−4.93i)T+(13.5−23.3i)T2 1 + (8.54 - 4.93i)T + (13.5 - 23.3i)T^{2}
5 1+(−3.19−1.84i)T+(62.5+108.i)T2 1 + (-3.19 - 1.84i)T + (62.5 + 108. i)T^{2}
11 1+(−28.5+16.4i)T+(665.5−1.15e3i)T2 1 + (-28.5 + 16.4i)T + (665.5 - 1.15e3i)T^{2}
13 1+33.0iT−2.19e3T2 1 + 33.0iT - 2.19e3T^{2}
17 1+(−26.6−46.2i)T+(−2.45e3+4.25e3i)T2 1 + (-26.6 - 46.2i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(44.3+25.6i)T+(3.42e3+5.94e3i)T2 1 + (44.3 + 25.6i)T + (3.42e3 + 5.94e3i)T^{2}
23 1+(−53.8+93.3i)T+(−6.08e3−1.05e4i)T2 1 + (-53.8 + 93.3i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1−7.61iT−2.43e4T2 1 - 7.61iT - 2.43e4T^{2}
31 1+(70.9+122.i)T+(−1.48e4+2.57e4i)T2 1 + (70.9 + 122. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(326.+188.i)T+(2.53e4+4.38e4i)T2 1 + (326. + 188. i)T + (2.53e4 + 4.38e4i)T^{2}
41 1+123.T+6.89e4T2 1 + 123.T + 6.89e4T^{2}
43 1−35.6iT−7.95e4T2 1 - 35.6iT - 7.95e4T^{2}
47 1+(−106.+184.i)T+(−5.19e4−8.99e4i)T2 1 + (-106. + 184. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−396.+228.i)T+(7.44e4−1.28e5i)T2 1 + (-396. + 228. i)T + (7.44e4 - 1.28e5i)T^{2}
59 1+(−70.4+40.6i)T+(1.02e5−1.77e5i)T2 1 + (-70.4 + 40.6i)T + (1.02e5 - 1.77e5i)T^{2}
61 1+(438.+252.i)T+(1.13e5+1.96e5i)T2 1 + (438. + 252. i)T + (1.13e5 + 1.96e5i)T^{2}
67 1+(171.−99.1i)T+(1.50e5−2.60e5i)T2 1 + (171. - 99.1i)T + (1.50e5 - 2.60e5i)T^{2}
71 1+474.T+3.57e5T2 1 + 474.T + 3.57e5T^{2}
73 1+(−337.−585.i)T+(−1.94e5+3.36e5i)T2 1 + (-337. - 585. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(−489.+847.i)T+(−2.46e5−4.26e5i)T2 1 + (-489. + 847. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+956.iT−5.71e5T2 1 + 956. iT - 5.71e5T^{2}
89 1+(512.−888.i)T+(−3.52e5−6.10e5i)T2 1 + (512. - 888. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+76.6T+9.12e5T2 1 + 76.6T + 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

−15.09140601013311169466927089795, −12.77788623136702507932556961104, −11.88311383674327301356601243116, −10.66612665529434969505268178180, −10.16718746275267033133156545569, −9.005387494553308716318605085568, −6.74587838784679713707367878837, −5.89510383200841257996269875628, −3.74077915843195954610131636156, −0.37824750320952361832080947525, 1.48577639387637544214727698095, 5.44793151025188811760219005016, 6.56085163442995046761030365631, 7.26168642335604768049640589211, 9.246902270721017101429967509985, 10.39764493332172290854519147990, 11.65362746095655200304883608679, 12.33640137232006527026118662732, 13.68614519954669990020535760135, 15.55834110048096833227156421682

Graph of the ZZ-function along the critical line