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Properties

Label 2-855-19.7-c1-0-1
Degree 22
Conductor 855855
Sign −0.0977+0.995i-0.0977 + 0.995i
Analytic cond. 6.827206.82720
Root an. cond. 2.612892.61289
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  + (1.20 + 2.09i)2-s + (−1.91 + 3.31i)4-s + (−0.5 − 0.866i)5-s − 3.82·7-s − 4.41·8-s + (1.20 − 2.09i)10-s − 2.82·11-s + (−1.91 + 3.31i)13-s + (−4.62 − 8.00i)14-s + (−1.49 − 2.59i)16-s + (−3.41 − 5.91i)17-s + (4 + 1.73i)19-s + 3.82·20-s + (−3.41 − 5.91i)22-s + (2.41 − 4.18i)23-s + ⋯
L(s)  = 1  + (0.853 + 1.47i)2-s + (−0.957 + 1.65i)4-s + (−0.223 − 0.387i)5-s − 1.44·7-s − 1.56·8-s + (0.381 − 0.661i)10-s − 0.852·11-s + (−0.530 + 0.919i)13-s + (−1.23 − 2.13i)14-s + (−0.374 − 0.649i)16-s + (−0.828 − 1.43i)17-s + (0.917 + 0.397i)19-s + 0.856·20-s + (−0.727 − 1.26i)22-s + (0.503 − 0.871i)23-s + ⋯

Functional equation

Λ(s)=(855s/2 ΓC(s) L(s)=((−0.0977+0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(855s/2 ΓC(s+1/2) L(s)=((−0.0977+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 855855    =    32⋅5⋅193^{2} \cdot 5 \cdot 19
Sign: −0.0977+0.995i-0.0977 + 0.995i
Analytic conductor: 6.827206.82720
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ855(406,⋅)\chi_{855} (406, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 855, ( :1/2), −0.0977+0.995i)(2,\ 855,\ (\ :1/2),\ -0.0977 + 0.995i)

Particular Values

L(1)L(1) ≈\approx 0.278820−0.307543i0.278820 - 0.307543i
L(12)L(\frac12) ≈\approx 0.278820−0.307543i0.278820 - 0.307543i
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)
bad3 1 1
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
19 1+(−4−1.73i)T 1 + (-4 - 1.73i)T
good2 1+(−1.20−2.09i)T+(−1+1.73i)T2 1 + (-1.20 - 2.09i)T + (-1 + 1.73i)T^{2}
7 1+3.82T+7T2 1 + 3.82T + 7T^{2}
11 1+2.82T+11T2 1 + 2.82T + 11T^{2}
13 1+(1.91−3.31i)T+(−6.5−11.2i)T2 1 + (1.91 - 3.31i)T + (-6.5 - 11.2i)T^{2}
17 1+(3.41+5.91i)T+(−8.5+14.7i)T2 1 + (3.41 + 5.91i)T + (-8.5 + 14.7i)T^{2}
23 1+(−2.41+4.18i)T+(−11.5−19.9i)T2 1 + (-2.41 + 4.18i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.828−1.43i)T+(−14.5−25.1i)T2 1 + (0.828 - 1.43i)T + (-14.5 - 25.1i)T^{2}
31 1+5T+31T2 1 + 5T + 31T^{2}
37 1+7.82T+37T2 1 + 7.82T + 37T^{2}
41 1+(1.41+2.44i)T+(−20.5+35.5i)T2 1 + (1.41 + 2.44i)T + (-20.5 + 35.5i)T^{2}
43 1+(1.08+1.88i)T+(−21.5+37.2i)T2 1 + (1.08 + 1.88i)T + (-21.5 + 37.2i)T^{2}
47 1+(4.41−7.64i)T+(−23.5−40.7i)T2 1 + (4.41 - 7.64i)T + (-23.5 - 40.7i)T^{2}
53 1+(1−1.73i)T+(−26.5−45.8i)T2 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2}
59 1+(−29.5+51.0i)T2 1 + (-29.5 + 51.0i)T^{2}
61 1+(7.15−12.3i)T+(−30.5−52.8i)T2 1 + (7.15 - 12.3i)T + (-30.5 - 52.8i)T^{2}
67 1+(−5.74+9.94i)T+(−33.5−58.0i)T2 1 + (-5.74 + 9.94i)T + (-33.5 - 58.0i)T^{2}
71 1+(−5−8.66i)T+(−35.5+61.4i)T2 1 + (-5 - 8.66i)T + (-35.5 + 61.4i)T^{2}
73 1+(−3.74−6.48i)T+(−36.5+63.2i)T2 1 + (-3.74 - 6.48i)T + (-36.5 + 63.2i)T^{2}
79 1+(−7.32−12.6i)T+(−39.5+68.4i)T2 1 + (-7.32 - 12.6i)T + (-39.5 + 68.4i)T^{2}
83 1−8T+83T2 1 - 8T + 83T^{2}
89 1+(2.24−3.88i)T+(−44.5−77.0i)T2 1 + (2.24 - 3.88i)T + (-44.5 - 77.0i)T^{2}
97 1+(−3−5.19i)T+(−48.5+84.0i)T2 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)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

−10.77768192069703597719020214192, −9.529725391583880066031135354326, −9.046892409740634023086598909624, −7.933425913025809496941666197780, −6.98427089782899625206182493341, −6.72688556460919425717174805658, −5.47604792205812285833211971385, −4.89282408308823625802978593287, −3.83131101300964818251869368282, −2.77931632868215756296614681487, 0.14091225595846314830869586172, 2.05070944851746593685893097010, 3.25065695249925714829790988553, 3.47933169873033164158730829236, 4.91284907153272873165746223580, 5.71111025177151462283641598065, 6.78177968712170771112649359570, 7.85655746134008778644626033773, 9.194408617558616987773922792443, 9.949191021937475986026007195661

Graph of the ZZ-function along the critical line