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Properties

Label 2-104-8.5-c1-0-6
Degree 22
Conductor 104104
Sign 0.707−0.707i0.707 - 0.707i
Analytic cond. 0.8304440.830444
Root an. cond. 0.9112870.911287
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  + (1.36 + 0.366i)2-s + 2i·3-s + (1.73 + i)4-s − 3.46i·5-s + (−0.732 + 2.73i)6-s − 4.73·7-s + (1.99 + 2i)8-s − 9-s + (1.26 − 4.73i)10-s − 1.26i·11-s + (−2 + 3.46i)12-s + i·13-s + (−6.46 − 1.73i)14-s + 6.92·15-s + (1.99 + 3.46i)16-s − 1.46·17-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + 1.15i·3-s + (0.866 + 0.5i)4-s − 1.54i·5-s + (−0.298 + 1.11i)6-s − 1.78·7-s + (0.707 + 0.707i)8-s − 0.333·9-s + (0.400 − 1.49i)10-s − 0.382i·11-s + (−0.577 + 0.999i)12-s + 0.277i·13-s + (−1.72 − 0.462i)14-s + 1.78·15-s + (0.499 + 0.866i)16-s − 0.355·17-s + ⋯

Functional equation

Λ(s)=(104s/2 ΓC(s) L(s)=((0.707−0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(104s/2 ΓC(s+1/2) L(s)=((0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 104104    =    23⋅132^{3} \cdot 13
Sign: 0.707−0.707i0.707 - 0.707i
Analytic conductor: 0.8304440.830444
Root analytic conductor: 0.9112870.911287
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ104(53,⋅)\chi_{104} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 104, ( :1/2), 0.707−0.707i)(2,\ 104,\ (\ :1/2),\ 0.707 - 0.707i)

Particular Values

L(1)L(1) ≈\approx 1.41988+0.588134i1.41988 + 0.588134i
L(12)L(\frac12) ≈\approx 1.41988+0.588134i1.41988 + 0.588134i
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+(−1.36−0.366i)T 1 + (-1.36 - 0.366i)T
13 1−iT 1 - iT
good3 1−2iT−3T2 1 - 2iT - 3T^{2}
5 1+3.46iT−5T2 1 + 3.46iT - 5T^{2}
7 1+4.73T+7T2 1 + 4.73T + 7T^{2}
11 1+1.26iT−11T2 1 + 1.26iT - 11T^{2}
17 1+1.46T+17T2 1 + 1.46T + 17T^{2}
19 1+2.73iT−19T2 1 + 2.73iT - 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1−2iT−29T2 1 - 2iT - 29T^{2}
31 1+3.26T+31T2 1 + 3.26T + 31T^{2}
37 1−4.92iT−37T2 1 - 4.92iT - 37T^{2}
41 1+4.92T+41T2 1 + 4.92T + 41T^{2}
43 1+7.46iT−43T2 1 + 7.46iT - 43T^{2}
47 1−3.26T+47T2 1 - 3.26T + 47T^{2}
53 1−10.9iT−53T2 1 - 10.9iT - 53T^{2}
59 1−0.196iT−59T2 1 - 0.196iT - 59T^{2}
61 1+10.9iT−61T2 1 + 10.9iT - 61T^{2}
67 1+2.73iT−67T2 1 + 2.73iT - 67T^{2}
71 1−2.19T+71T2 1 - 2.19T + 71T^{2}
73 1+0.535T+73T2 1 + 0.535T + 73T^{2}
79 1+1.46T+79T2 1 + 1.46T + 79T^{2}
83 1−6.73iT−83T2 1 - 6.73iT - 83T^{2}
89 1−17.3T+89T2 1 - 17.3T + 89T^{2}
97 1+14.3T+97T2 1 + 14.3T + 97T^{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

−13.66713348460551342477738804601, −12.97108562948306335020693325602, −12.21653871489260166528855857102, −10.78546377761365198318805053420, −9.514710332505391917145492678906, −8.769027114441600982185064524032, −6.85944620926091183156489416071, −5.50205270731119407441896095195, −4.48177866106527543337411071211, −3.35667813713256326653348392359, 2.48106307219566836217102578034, 3.56327089894437307225813162307, 6.03341198059096965809599785503, 6.75796848113682246252332617157, 7.37029897698755686362765248658, 9.789178435230406437411541607628, 10.64765648977790656680617448684, 11.88890447864364030441797403380, 12.88038802413954090975969281922, 13.36245056849728059608579069507

Graph of the ZZ-function along the critical line