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Properties

Label 2-104-8.5-c1-0-1
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  + (−0.366 − 1.36i)2-s + 2i·3-s + (−1.73 + i)4-s + 3.46i·5-s + (2.73 − 0.732i)6-s − 1.26·7-s + (2 + 1.99i)8-s − 9-s + (4.73 − 1.26i)10-s − 4.73i·11-s + (−2 − 3.46i)12-s + i·13-s + (0.464 + 1.73i)14-s − 6.92·15-s + (1.99 − 3.46i)16-s + 5.46·17-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + 1.15i·3-s + (−0.866 + 0.5i)4-s + 1.54i·5-s + (1.11 − 0.298i)6-s − 0.479·7-s + (0.707 + 0.707i)8-s − 0.333·9-s + (1.49 − 0.400i)10-s − 1.42i·11-s + (−0.577 − 0.999i)12-s + 0.277i·13-s + (0.124 + 0.462i)14-s − 1.78·15-s + (0.499 − 0.866i)16-s + 1.32·17-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 0.756322+0.313279i0.756322 + 0.313279i
L(12)L(\frac12) ≈\approx 0.756322+0.313279i0.756322 + 0.313279i
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+(0.366+1.36i)T 1 + (0.366 + 1.36i)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+1.26T+7T2 1 + 1.26T + 7T^{2}
11 1+4.73iT−11T2 1 + 4.73iT - 11T^{2}
17 1−5.46T+17T2 1 - 5.46T + 17T^{2}
19 1−0.732iT−19T2 1 - 0.732iT - 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1−2iT−29T2 1 - 2iT - 29T^{2}
31 1+6.73T+31T2 1 + 6.73T + 31T^{2}
37 1+8.92iT−37T2 1 + 8.92iT - 37T^{2}
41 1−8.92T+41T2 1 - 8.92T + 41T^{2}
43 1+0.535iT−43T2 1 + 0.535iT - 43T^{2}
47 1−6.73T+47T2 1 - 6.73T + 47T^{2}
53 1+2.92iT−53T2 1 + 2.92iT - 53T^{2}
59 1+10.1iT−59T2 1 + 10.1iT - 59T^{2}
61 1−2.92iT−61T2 1 - 2.92iT - 61T^{2}
67 1−0.732iT−67T2 1 - 0.732iT - 67T^{2}
71 1+8.19T+71T2 1 + 8.19T + 71T^{2}
73 1+7.46T+73T2 1 + 7.46T + 73T^{2}
79 1−5.46T+79T2 1 - 5.46T + 79T^{2}
83 1−3.26iT−83T2 1 - 3.26iT - 83T^{2}
89 1+17.3T+89T2 1 + 17.3T + 89T^{2}
97 1−6.39T+97T2 1 - 6.39T + 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

−14.13138761130749281911969637587, −12.74559147605110047710515783750, −11.22936593423537180748242896467, −10.80347344142736103279795786413, −9.927458358212439716120657493433, −8.999722199462404836567638118216, −7.39930033665639993270289510149, −5.65706916745960590564750799286, −3.75203968154316350580012571200, −3.06320050117788563455232785257, 1.23000020128735635730991750156, 4.56746430097494365243897227587, 5.76050998433423880724887136900, 7.15368383405723015950202175585, 7.88710400559734548488988068574, 9.093944912592433757917183976042, 9.975598714293025556706726442622, 12.22471881712449308840253616255, 12.77478134770588876714157957757, 13.41041458260834501861777971835

Graph of the ZZ-function along the critical line