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Properties

Label 2-104-104.83-c1-0-8
Degree 22
Conductor 104104
Sign 0.829−0.558i0.829 - 0.558i
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  + (1.15 + 0.813i)2-s + 1.38·3-s + (0.674 + 1.88i)4-s + (−2.40 − 2.40i)5-s + (1.60 + 1.12i)6-s + (−0.127 + 0.127i)7-s + (−0.751 + 2.72i)8-s − 1.07·9-s + (−0.825 − 4.74i)10-s + (−2.63 − 2.63i)11-s + (0.936 + 2.61i)12-s + (1.93 + 3.04i)13-s + (−0.250 + 0.0435i)14-s + (−3.34 − 3.34i)15-s + (−3.08 + 2.54i)16-s + 4.33i·17-s + ⋯
L(s)  = 1  + (0.817 + 0.575i)2-s + 0.801·3-s + (0.337 + 0.941i)4-s + (−1.07 − 1.07i)5-s + (0.655 + 0.461i)6-s + (−0.0480 + 0.0480i)7-s + (−0.265 + 0.964i)8-s − 0.357·9-s + (−0.260 − 1.50i)10-s + (−0.793 − 0.793i)11-s + (0.270 + 0.754i)12-s + (0.537 + 0.843i)13-s + (−0.0668 + 0.0116i)14-s + (−0.863 − 0.863i)15-s + (−0.772 + 0.635i)16-s + 1.05i·17-s + ⋯

Functional equation

Λ(s)=(104s/2 ΓC(s) L(s)=((0.829−0.558i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(104s/2 ΓC(s+1/2) L(s)=((0.829−0.558i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 104104    =    23⋅132^{3} \cdot 13
Sign: 0.829−0.558i0.829 - 0.558i
Analytic conductor: 0.8304440.830444
Root analytic conductor: 0.9112870.911287
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ104(83,⋅)\chi_{104} (83, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 104, ( :1/2), 0.829−0.558i)(2,\ 104,\ (\ :1/2),\ 0.829 - 0.558i)

Particular Values

L(1)L(1) ≈\approx 1.50434+0.459233i1.50434 + 0.459233i
L(12)L(\frac12) ≈\approx 1.50434+0.459233i1.50434 + 0.459233i
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+(−1.15−0.813i)T 1 + (-1.15 - 0.813i)T
13 1+(−1.93−3.04i)T 1 + (-1.93 - 3.04i)T
good3 1−1.38T+3T2 1 - 1.38T + 3T^{2}
5 1+(2.40+2.40i)T+5iT2 1 + (2.40 + 2.40i)T + 5iT^{2}
7 1+(0.127−0.127i)T−7iT2 1 + (0.127 - 0.127i)T - 7iT^{2}
11 1+(2.63+2.63i)T+11iT2 1 + (2.63 + 2.63i)T + 11iT^{2}
17 1−4.33iT−17T2 1 - 4.33iT - 17T^{2}
19 1+(−4.97+4.97i)T−19iT2 1 + (-4.97 + 4.97i)T - 19iT^{2}
23 1−3.98T+23T2 1 - 3.98T + 23T^{2}
29 1+4.59iT−29T2 1 + 4.59iT - 29T^{2}
31 1+(−1.07−1.07i)T+31iT2 1 + (-1.07 - 1.07i)T + 31iT^{2}
37 1+(2.45−2.45i)T−37iT2 1 + (2.45 - 2.45i)T - 37iT^{2}
41 1+(−0.388+0.388i)T−41iT2 1 + (-0.388 + 0.388i)T - 41iT^{2}
43 1+5.02iT−43T2 1 + 5.02iT - 43T^{2}
47 1+(−1.00+1.00i)T−47iT2 1 + (-1.00 + 1.00i)T - 47iT^{2}
53 1+1.83iT−53T2 1 + 1.83iT - 53T^{2}
59 1+(−6.28−6.28i)T+59iT2 1 + (-6.28 - 6.28i)T + 59iT^{2}
61 1−10.5iT−61T2 1 - 10.5iT - 61T^{2}
67 1+(4.21−4.21i)T−67iT2 1 + (4.21 - 4.21i)T - 67iT^{2}
71 1+(2.59+2.59i)T+71iT2 1 + (2.59 + 2.59i)T + 71iT^{2}
73 1+(0.388+0.388i)T+73iT2 1 + (0.388 + 0.388i)T + 73iT^{2}
79 1+15.3iT−79T2 1 + 15.3iT - 79T^{2}
83 1+(2.46−2.46i)T−83iT2 1 + (2.46 - 2.46i)T - 83iT^{2}
89 1+(11.3+11.3i)T+89iT2 1 + (11.3 + 11.3i)T + 89iT^{2}
97 1+(−3.31+3.31i)T−97iT2 1 + (-3.31 + 3.31i)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

−13.71331418011072538241207872032, −13.21933553830647714063162284847, −11.97733926843294803718997268531, −11.19368962892920551559682897028, −8.869668120273623231966486408851, −8.432287381874690846597059988580, −7.37571114040786143428908254357, −5.65347815859101825072903709678, −4.33948708867299266456859597164, −3.11792598352996510764840247593, 2.81960985573929096596803077258, 3.56044134354854554941063967510, 5.31427496746830413111692945808, 7.05762134595474572748386341933, 7.981637153956632485227531968440, 9.677074953147536704181633494198, 10.76525484983058202798160627459, 11.59238934718402845552404258499, 12.68454633718648644487177692331, 13.84102658207858165518266712765

Graph of the ZZ-function along the critical line