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Properties

Label 2-855-15.8-c1-0-22
Degree 22
Conductor 855855
Sign −0.990−0.139i-0.990 - 0.139i
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  + (−1.41 − 1.41i)2-s + 2.00i·4-s + (0.448 − 2.19i)5-s + (2.36 − 2.36i)7-s + (−3.73 + 2.46i)10-s − 1.55i·11-s + (0.732 + 0.732i)13-s − 6.69·14-s + 3.99·16-s + (−4.57 − 4.57i)17-s + i·19-s + (4.38 + 0.896i)20-s + (−2.19 + 2.19i)22-s + (2.44 − 2.44i)23-s + (−4.59 − 1.96i)25-s − 2.07i·26-s + ⋯
L(s)  = 1  + (−0.999 − 0.999i)2-s + 1.00i·4-s + (0.200 − 0.979i)5-s + (0.894 − 0.894i)7-s + (−1.18 + 0.779i)10-s − 0.468i·11-s + (0.203 + 0.203i)13-s − 1.78·14-s + 0.999·16-s + (−1.10 − 1.10i)17-s + 0.229i·19-s + (0.979 + 0.200i)20-s + (−0.468 + 0.468i)22-s + (0.510 − 0.510i)23-s + (−0.919 − 0.392i)25-s − 0.406i·26-s + ⋯

Functional equation

Λ(s)=(855s/2 ΓC(s) L(s)=((−0.990−0.139i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(855s/2 ΓC(s+1/2) L(s)=((−0.990−0.139i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 855855    =    32⋅5⋅193^{2} \cdot 5 \cdot 19
Sign: −0.990−0.139i-0.990 - 0.139i
Analytic conductor: 6.827206.82720
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ855(818,⋅)\chi_{855} (818, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 855, ( :1/2), −0.990−0.139i)(2,\ 855,\ (\ :1/2),\ -0.990 - 0.139i)

Particular Values

L(1)L(1) ≈\approx 0.0603121+0.860547i0.0603121 + 0.860547i
L(12)L(\frac12) ≈\approx 0.0603121+0.860547i0.0603121 + 0.860547i
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+(−0.448+2.19i)T 1 + (-0.448 + 2.19i)T
19 1−iT 1 - iT
good2 1+(1.41+1.41i)T+2iT2 1 + (1.41 + 1.41i)T + 2iT^{2}
7 1+(−2.36+2.36i)T−7iT2 1 + (-2.36 + 2.36i)T - 7iT^{2}
11 1+1.55iT−11T2 1 + 1.55iT - 11T^{2}
13 1+(−0.732−0.732i)T+13iT2 1 + (-0.732 - 0.732i)T + 13iT^{2}
17 1+(4.57+4.57i)T+17iT2 1 + (4.57 + 4.57i)T + 17iT^{2}
23 1+(−2.44+2.44i)T−23iT2 1 + (-2.44 + 2.44i)T - 23iT^{2}
29 1−8.76T+29T2 1 - 8.76T + 29T^{2}
31 1+0.535T+31T2 1 + 0.535T + 31T^{2}
37 1+(−3.46+3.46i)T−37iT2 1 + (-3.46 + 3.46i)T - 37iT^{2}
41 1−2.82iT−41T2 1 - 2.82iT - 41T^{2}
43 1+(3.09+3.09i)T+43iT2 1 + (3.09 + 3.09i)T + 43iT^{2}
47 1+(−2.63−2.63i)T+47iT2 1 + (-2.63 - 2.63i)T + 47iT^{2}
53 1+(4.24−4.24i)T−53iT2 1 + (4.24 - 4.24i)T - 53iT^{2}
59 1+13.3T+59T2 1 + 13.3T + 59T^{2}
61 1+12.8T+61T2 1 + 12.8T + 61T^{2}
67 1+(−9.46+9.46i)T−67iT2 1 + (-9.46 + 9.46i)T - 67iT^{2}
71 1−0.757iT−71T2 1 - 0.757iT - 71T^{2}
73 1+(3.90+3.90i)T+73iT2 1 + (3.90 + 3.90i)T + 73iT^{2}
79 1−13.4iT−79T2 1 - 13.4iT - 79T^{2}
83 1+(−0.757+0.757i)T−83iT2 1 + (-0.757 + 0.757i)T - 83iT^{2}
89 1+8.76T+89T2 1 + 8.76T + 89T^{2}
97 1+(5.66−5.66i)T−97iT2 1 + (5.66 - 5.66i)T - 97iT^{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

−9.670602408605097508814834804380, −9.016565546564050166312399566787, −8.353070905662392134554706592801, −7.61519269112524189424366122894, −6.32958960787657719250993097572, −5.00607144157119136609480061433, −4.29005231560899561709172934701, −2.79483570346346873749141457088, −1.54630250482827433975175338122, −0.61695294130644760877765392275, 1.75712474657630524395957549895, 3.04069773862045940234842816532, 4.59969046645853080348702543714, 5.80896160556610727981514315156, 6.46860100926071715488518746017, 7.25505024324068241861885216674, 8.171088489835134669853109195254, 8.692329712559667251491492095619, 9.586973198648706763450935511138, 10.43723070647169722056854556080

Graph of the ZZ-function along the critical line