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Properties

Label 2-55-11.6-c2-0-5
Degree 22
Conductor 5555
Sign 0.918+0.396i0.918 + 0.396i
Analytic cond. 1.498641.49864
Root an. cond. 1.224191.22419
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.68 − 0.872i)2-s + (0.0862 + 0.0626i)3-s + (3.21 − 2.33i)4-s + (−0.690 + 2.12i)5-s + (0.286 + 0.0929i)6-s + (−1.76 − 2.42i)7-s + (−0.0523 + 0.0721i)8-s + (−2.77 − 8.54i)9-s + 6.31i·10-s + (−4.57 + 10.0i)11-s + 0.423·12-s + (0.925 − 0.300i)13-s + (−6.85 − 4.98i)14-s + (−0.192 + 0.140i)15-s + (−4.98 + 15.3i)16-s + (4.85 + 1.57i)17-s + ⋯
L(s)  = 1  + (1.34 − 0.436i)2-s + (0.0287 + 0.0208i)3-s + (0.802 − 0.583i)4-s + (−0.138 + 0.425i)5-s + (0.0476 + 0.0154i)6-s + (−0.252 − 0.347i)7-s + (−0.00654 + 0.00901i)8-s + (−0.308 − 0.949i)9-s + 0.631i·10-s + (−0.416 + 0.909i)11-s + 0.0352·12-s + (0.0711 − 0.0231i)13-s + (−0.489 − 0.355i)14-s + (−0.0128 + 0.00934i)15-s + (−0.311 + 0.958i)16-s + (0.285 + 0.0928i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.918+0.396i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1) L(s)=((0.918+0.396i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.918+0.396i0.918 + 0.396i
Analytic conductor: 1.498641.49864
Root analytic conductor: 1.224191.22419
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ55(6,⋅)\chi_{55} (6, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1), 0.918+0.396i)(2,\ 55,\ (\ :1),\ 0.918 + 0.396i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.94267−0.401101i1.94267 - 0.401101i
L(12)L(\frac12) ≈\approx 1.94267−0.401101i1.94267 - 0.401101i
L(2)L(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)
bad5 1+(0.690−2.12i)T 1 + (0.690 - 2.12i)T
11 1+(4.57−10.0i)T 1 + (4.57 - 10.0i)T
good2 1+(−2.68+0.872i)T+(3.23−2.35i)T2 1 + (-2.68 + 0.872i)T + (3.23 - 2.35i)T^{2}
3 1+(−0.0862−0.0626i)T+(2.78+8.55i)T2 1 + (-0.0862 - 0.0626i)T + (2.78 + 8.55i)T^{2}
7 1+(1.76+2.42i)T+(−15.1+46.6i)T2 1 + (1.76 + 2.42i)T + (-15.1 + 46.6i)T^{2}
13 1+(−0.925+0.300i)T+(136.−99.3i)T2 1 + (-0.925 + 0.300i)T + (136. - 99.3i)T^{2}
17 1+(−4.85−1.57i)T+(233.+169.i)T2 1 + (-4.85 - 1.57i)T + (233. + 169. i)T^{2}
19 1+(−7.85+10.8i)T+(−111.−343.i)T2 1 + (-7.85 + 10.8i)T + (-111. - 343. i)T^{2}
23 1−24.5T+529T2 1 - 24.5T + 529T^{2}
29 1+(19.2+26.5i)T+(−259.+799.i)T2 1 + (19.2 + 26.5i)T + (-259. + 799. i)T^{2}
31 1+(3.40+10.4i)T+(−777.+564.i)T2 1 + (3.40 + 10.4i)T + (-777. + 564. i)T^{2}
37 1+(7.65−5.56i)T+(423.−1.30e3i)T2 1 + (7.65 - 5.56i)T + (423. - 1.30e3i)T^{2}
41 1+(−43.5+59.9i)T+(−519.−1.59e3i)T2 1 + (-43.5 + 59.9i)T + (-519. - 1.59e3i)T^{2}
43 1−52.1iT−1.84e3T2 1 - 52.1iT - 1.84e3T^{2}
47 1+(−33.4−24.2i)T+(682.+2.10e3i)T2 1 + (-33.4 - 24.2i)T + (682. + 2.10e3i)T^{2}
53 1+(−19.3−59.4i)T+(−2.27e3+1.65e3i)T2 1 + (-19.3 - 59.4i)T + (-2.27e3 + 1.65e3i)T^{2}
59 1+(57.2−41.5i)T+(1.07e3−3.31e3i)T2 1 + (57.2 - 41.5i)T + (1.07e3 - 3.31e3i)T^{2}
61 1+(69.3+22.5i)T+(3.01e3+2.18e3i)T2 1 + (69.3 + 22.5i)T + (3.01e3 + 2.18e3i)T^{2}
67 1+116.T+4.48e3T2 1 + 116.T + 4.48e3T^{2}
71 1+(38.9−119.i)T+(−4.07e3−2.96e3i)T2 1 + (38.9 - 119. i)T + (-4.07e3 - 2.96e3i)T^{2}
73 1+(11.6+16.0i)T+(−1.64e3+5.06e3i)T2 1 + (11.6 + 16.0i)T + (-1.64e3 + 5.06e3i)T^{2}
79 1+(−31.0+10.1i)T+(5.04e3−3.66e3i)T2 1 + (-31.0 + 10.1i)T + (5.04e3 - 3.66e3i)T^{2}
83 1+(−138.−44.9i)T+(5.57e3+4.04e3i)T2 1 + (-138. - 44.9i)T + (5.57e3 + 4.04e3i)T^{2}
89 1+94.7T+7.92e3T2 1 + 94.7T + 7.92e3T^{2}
97 1+(42.4+130.i)T+(−7.61e3+5.53e3i)T2 1 + (42.4 + 130. i)T + (-7.61e3 + 5.53e3i)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

−14.88508264713138928928529879345, −13.77968629157741768673247330860, −12.77587425086690289790081635571, −11.86118928720529937392493138104, −10.74395106275120031756254568431, −9.263255174924732416231533400933, −7.27933255900320559353014299325, −5.85697510858194832709784940879, −4.27501513238284849703587537622, −2.91575294438019046833768318308, 3.20519655280078667862854042591, 4.97712269824737475144473752257, 5.88650060530106413442106181285, 7.57144632210813432911226398510, 9.025464779849471229055407762960, 10.83722752289923113190338068657, 12.13579305330249568051728984930, 13.15418370192447636378238968768, 13.89814204294550359394449360438, 14.98092410464650716682704554958

Graph of the ZZ-function along the critical line