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Properties

Label 2-10e3-1000.109-c1-0-100
Degree 22
Conductor 10001000
Sign 0.999+0.0350i0.999 + 0.0350i
Analytic cond. 7.985047.98504
Root an. cond. 2.825782.82578
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  + (−0.894 + 1.09i)2-s + (0.859 + 1.03i)3-s + (−0.399 − 1.95i)4-s + (0.254 − 2.22i)5-s + (−1.90 + 0.0118i)6-s + (4.65 − 1.51i)7-s + (2.50 + 1.31i)8-s + (0.221 − 1.16i)9-s + (2.20 + 2.26i)10-s + (−0.350 + 2.77i)11-s + (1.69 − 2.09i)12-s + (−0.189 + 0.995i)13-s + (−2.50 + 6.45i)14-s + (2.52 − 1.64i)15-s + (−3.68 + 1.56i)16-s + (−1.66 − 6.48i)17-s + ⋯
L(s)  = 1  + (−0.632 + 0.774i)2-s + (0.496 + 0.599i)3-s + (−0.199 − 0.979i)4-s + (0.113 − 0.993i)5-s + (−0.778 + 0.00482i)6-s + (1.75 − 0.571i)7-s + (0.885 + 0.465i)8-s + (0.0738 − 0.386i)9-s + (0.697 + 0.716i)10-s + (−0.105 + 0.837i)11-s + (0.488 − 0.605i)12-s + (−0.0526 + 0.276i)13-s + (−0.670 + 1.72i)14-s + (0.652 − 0.424i)15-s + (−0.920 + 0.391i)16-s + (−0.403 − 1.57i)17-s + ⋯

Functional equation

Λ(s)=(1000s/2 ΓC(s) L(s)=((0.999+0.0350i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0350i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1000s/2 ΓC(s+1/2) L(s)=((0.999+0.0350i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0350i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10001000    =    23⋅532^{3} \cdot 5^{3}
Sign: 0.999+0.0350i0.999 + 0.0350i
Analytic conductor: 7.985047.98504
Root analytic conductor: 2.825782.82578
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1000(109,⋅)\chi_{1000} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1000, ( :1/2), 0.999+0.0350i)(2,\ 1000,\ (\ :1/2),\ 0.999 + 0.0350i)

Particular Values

L(1)L(1) ≈\approx 1.60952−0.0281835i1.60952 - 0.0281835i
L(12)L(\frac12) ≈\approx 1.60952−0.0281835i1.60952 - 0.0281835i
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)
bad2 1+(0.894−1.09i)T 1 + (0.894 - 1.09i)T
5 1+(−0.254+2.22i)T 1 + (-0.254 + 2.22i)T
good3 1+(−0.859−1.03i)T+(−0.562+2.94i)T2 1 + (-0.859 - 1.03i)T + (-0.562 + 2.94i)T^{2}
7 1+(−4.65+1.51i)T+(5.66−4.11i)T2 1 + (-4.65 + 1.51i)T + (5.66 - 4.11i)T^{2}
11 1+(0.350−2.77i)T+(−10.6−2.73i)T2 1 + (0.350 - 2.77i)T + (-10.6 - 2.73i)T^{2}
13 1+(0.189−0.995i)T+(−12.0−4.78i)T2 1 + (0.189 - 0.995i)T + (-12.0 - 4.78i)T^{2}
17 1+(1.66+6.48i)T+(−14.8+8.18i)T2 1 + (1.66 + 6.48i)T + (-14.8 + 8.18i)T^{2}
19 1+(1.29+1.06i)T+(3.56+18.6i)T2 1 + (1.29 + 1.06i)T + (3.56 + 18.6i)T^{2}
23 1+(−0.102+0.109i)T+(−1.44−22.9i)T2 1 + (-0.102 + 0.109i)T + (-1.44 - 22.9i)T^{2}
29 1+(2.65−0.166i)T+(28.7−3.63i)T2 1 + (2.65 - 0.166i)T + (28.7 - 3.63i)T^{2}
31 1+(−4.75+1.22i)T+(27.1−14.9i)T2 1 + (-4.75 + 1.22i)T + (27.1 - 14.9i)T^{2}
37 1+(4.28+2.35i)T+(19.8+31.2i)T2 1 + (4.28 + 2.35i)T + (19.8 + 31.2i)T^{2}
41 1+(−2.05+1.92i)T+(2.57−40.9i)T2 1 + (-2.05 + 1.92i)T + (2.57 - 40.9i)T^{2}
43 1+(−0.729+0.529i)T+(13.2−40.8i)T2 1 + (-0.729 + 0.529i)T + (13.2 - 40.8i)T^{2}
47 1+(−2.32−5.86i)T+(−34.2+32.1i)T2 1 + (-2.32 - 5.86i)T + (-34.2 + 32.1i)T^{2}
53 1+(4.68−7.37i)T+(−22.5−47.9i)T2 1 + (4.68 - 7.37i)T + (-22.5 - 47.9i)T^{2}
59 1+(3.28−1.54i)T+(37.6−45.4i)T2 1 + (3.28 - 1.54i)T + (37.6 - 45.4i)T^{2}
61 1+(−8.01+8.53i)T+(−3.83−60.8i)T2 1 + (-8.01 + 8.53i)T + (-3.83 - 60.8i)T^{2}
67 1+(0.269−4.27i)T+(−66.4−8.39i)T2 1 + (0.269 - 4.27i)T + (-66.4 - 8.39i)T^{2}
71 1+(8.35−3.30i)T+(51.7−48.6i)T2 1 + (8.35 - 3.30i)T + (51.7 - 48.6i)T^{2}
73 1+(0.209+0.0984i)T+(46.5+56.2i)T2 1 + (0.209 + 0.0984i)T + (46.5 + 56.2i)T^{2}
79 1+(−4.41−5.33i)T+(−14.8+77.6i)T2 1 + (-4.41 - 5.33i)T + (-14.8 + 77.6i)T^{2}
83 1+(0.0671−0.0811i)T+(−15.5−81.5i)T2 1 + (0.0671 - 0.0811i)T + (-15.5 - 81.5i)T^{2}
89 1+(5.81−12.3i)T+(−56.7−68.5i)T2 1 + (5.81 - 12.3i)T + (-56.7 - 68.5i)T^{2}
97 1+(−11.8+0.743i)T+(96.2−12.1i)T2 1 + (-11.8 + 0.743i)T + (96.2 - 12.1i)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

−9.586273356899496093983732912702, −9.135037464112426817099613166505, −8.368794017316765734363805640754, −7.63655284803034228316778805218, −6.88856692805358788429371526562, −5.46280108679405126917065761838, −4.59491535448105310061534860779, −4.36464534384649983046791270762, −2.13730914081516315781639352013, −0.939583917505518842557711162521, 1.61421711291225763285085646751, 2.20582553210932604048875926888, 3.26401061909585597422951562351, 4.48242389164259263914734434141, 5.72498508256833211947101499049, 6.91912520705651337953677382986, 7.983701373388142729304316246938, 8.169209487845162190156600317937, 8.897431801594798435173807243941, 10.33009640455115450966405854074

Graph of the ZZ-function along the critical line