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Properties

Label 2-104-8.5-c1-0-3
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 + i)2-s − i·3-s − 2i·4-s + 3i·5-s + (1 + i)6-s + 3·7-s + (2 + 2i)8-s + 2·9-s + (−3 − 3i)10-s − 2·12-s + i·13-s + (−3 + 3i)14-s + 3·15-s − 4·16-s − 7·17-s + (−2 + 2i)18-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 0.577i·3-s − i·4-s + 1.34i·5-s + (0.408 + 0.408i)6-s + 1.13·7-s + (0.707 + 0.707i)8-s + 0.666·9-s + (−0.948 − 0.948i)10-s − 0.577·12-s + 0.277i·13-s + (−0.801 + 0.801i)14-s + 0.774·15-s − 16-s − 1.69·17-s + (−0.471 + 0.471i)18-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 0.752271+0.311601i0.752271 + 0.311601i
L(12)L(\frac12) ≈\approx 0.752271+0.311601i0.752271 + 0.311601i
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−i)T 1 + (1 - i)T
13 1−iT 1 - iT
good3 1+iT−3T2 1 + iT - 3T^{2}
5 1−3iT−5T2 1 - 3iT - 5T^{2}
7 1−3T+7T2 1 - 3T + 7T^{2}
11 1−11T2 1 - 11T^{2}
17 1+7T+17T2 1 + 7T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1+4iT−29T2 1 + 4iT - 29T^{2}
31 1+8T+31T2 1 + 8T + 31T^{2}
37 1−7iT−37T2 1 - 7iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+iT−43T2 1 + iT - 43T^{2}
47 1+7T+47T2 1 + 7T + 47T^{2}
53 1−4iT−53T2 1 - 4iT - 53T^{2}
59 1+14iT−59T2 1 + 14iT - 59T^{2}
61 1+10iT−61T2 1 + 10iT - 61T^{2}
67 1−2iT−67T2 1 - 2iT - 67T^{2}
71 1+3T+71T2 1 + 3T + 71T^{2}
73 1−14T+73T2 1 - 14T + 73T^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1−14iT−83T2 1 - 14iT - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1−8T+97T2 1 - 8T + 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

−14.15440631000278160403879527824, −13.20852039438558746320617634653, −11.26399418836677676300121220421, −10.92856591718551450502520099306, −9.508812300858634453042338551359, −8.195138470375305827482347337470, −7.10015539971666652880095457705, −6.57035023944590742165974568824, −4.73879432561462994763240805629, −2.05759140072052388639569224374, 1.61690266206879990138128123661, 4.10530874863142447259873660113, 5.01634651797727751766763907520, 7.40053747850704823055861792041, 8.626171664997409426871552911051, 9.199662015815668273650387336959, 10.52597930620038220776880637397, 11.34323865242704371941958808037, 12.58600300582480593983015104864, 13.19648486565889657619060078782

Graph of the ZZ-function along the critical line