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Properties

Label 2-975-195.77-c1-0-2
Degree 22
Conductor 975975
Sign −0.924+0.379i-0.924 + 0.379i
Analytic cond. 7.785417.78541
Root an. cond. 2.790232.79023
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  + (0.299 + 0.299i)2-s + (0.125 + 1.72i)3-s − 1.82i·4-s + (−0.479 + 0.554i)6-s + (−0.976 + 0.976i)7-s + (1.14 − 1.14i)8-s + (−2.96 + 0.431i)9-s − 1.78·11-s + (3.14 − 0.227i)12-s + (−2.75 − 2.32i)13-s − 0.584·14-s − 2.95·16-s + (−3.26 + 3.26i)17-s + (−1.01 − 0.759i)18-s − 6.18·19-s + ⋯
L(s)  = 1  + (0.211 + 0.211i)2-s + (0.0721 + 0.997i)3-s − 0.910i·4-s + (−0.195 + 0.226i)6-s + (−0.368 + 0.368i)7-s + (0.404 − 0.404i)8-s + (−0.989 + 0.143i)9-s − 0.537·11-s + (0.908 − 0.0657i)12-s + (−0.765 − 0.643i)13-s − 0.156·14-s − 0.739·16-s + (−0.790 + 0.790i)17-s + (−0.239 − 0.178i)18-s − 1.41·19-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((−0.924+0.379i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s+1/2) L(s)=((−0.924+0.379i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: −0.924+0.379i-0.924 + 0.379i
Analytic conductor: 7.785417.78541
Root analytic conductor: 2.790232.79023
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ975(857,⋅)\chi_{975} (857, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :1/2), −0.924+0.379i)(2,\ 975,\ (\ :1/2),\ -0.924 + 0.379i)

Particular Values

L(1)L(1) ≈\approx 0.0301231−0.152601i0.0301231 - 0.152601i
L(12)L(\frac12) ≈\approx 0.0301231−0.152601i0.0301231 - 0.152601i
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+(−0.125−1.72i)T 1 + (-0.125 - 1.72i)T
5 1 1
13 1+(2.75+2.32i)T 1 + (2.75 + 2.32i)T
good2 1+(−0.299−0.299i)T+2iT2 1 + (-0.299 - 0.299i)T + 2iT^{2}
7 1+(0.976−0.976i)T−7iT2 1 + (0.976 - 0.976i)T - 7iT^{2}
11 1+1.78T+11T2 1 + 1.78T + 11T^{2}
17 1+(3.26−3.26i)T−17iT2 1 + (3.26 - 3.26i)T - 17iT^{2}
19 1+6.18T+19T2 1 + 6.18T + 19T^{2}
23 1+(−0.696−0.696i)T+23iT2 1 + (-0.696 - 0.696i)T + 23iT^{2}
29 1−7.33T+29T2 1 - 7.33T + 29T^{2}
31 1−6.61iT−31T2 1 - 6.61iT - 31T^{2}
37 1+(5.21−5.21i)T−37iT2 1 + (5.21 - 5.21i)T - 37iT^{2}
41 1+6.45T+41T2 1 + 6.45T + 41T^{2}
43 1+(−1.78+1.78i)T−43iT2 1 + (-1.78 + 1.78i)T - 43iT^{2}
47 1+(5.69+5.69i)T+47iT2 1 + (5.69 + 5.69i)T + 47iT^{2}
53 1+(1.74+1.74i)T+53iT2 1 + (1.74 + 1.74i)T + 53iT^{2}
59 1+9.15iT−59T2 1 + 9.15iT - 59T^{2}
61 1−1.01T+61T2 1 - 1.01T + 61T^{2}
67 1+(3.72−3.72i)T−67iT2 1 + (3.72 - 3.72i)T - 67iT^{2}
71 1−5.49T+71T2 1 - 5.49T + 71T^{2}
73 1+(−1.35−1.35i)T+73iT2 1 + (-1.35 - 1.35i)T + 73iT^{2}
79 1+10.2iT−79T2 1 + 10.2iT - 79T^{2}
83 1+(−7.93+7.93i)T−83iT2 1 + (-7.93 + 7.93i)T - 83iT^{2}
89 1+3.75iT−89T2 1 + 3.75iT - 89T^{2}
97 1+(−6.92+6.92i)T−97iT2 1 + (-6.92 + 6.92i)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

−10.40799302059570334526070132607, −9.878865175854569004376088822480, −8.838117170834551897596530996370, −8.269094243413640020637446418958, −6.77841705318171533120909168716, −6.11328285141204490137506352101, −5.08110115131388128524382191600, −4.63513575908265847467316704220, −3.31364463988422677234740759802, −2.14765681512664996195815228836, 0.05940032456887405762153628382, 2.14242524685791498467689178742, 2.80366030495277644985307621937, 4.08815363508976846330322353165, 5.01295494918612545981721917550, 6.48627682797197680524953120470, 6.96388705097467554354472860401, 7.81718844720653978241947460123, 8.536588287572663786081891952781, 9.354867719373889534775941414254

Graph of the ZZ-function along the critical line