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Properties

Label 2-104-104.69-c1-0-7
Degree 22
Conductor 104104
Sign 0.494+0.869i0.494 + 0.869i
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 + (0.633 + 0.366i)3-s + (−1.73 − i)4-s + 3.73·5-s + (0.732 − 0.732i)6-s + (−3 + 1.73i)7-s + (−2 + 1.99i)8-s + (−1.23 − 2.13i)9-s + (1.36 − 5.09i)10-s + (1 − 1.73i)11-s + (−0.732 − 1.26i)12-s + (−2.59 + 2.5i)13-s + (1.26 + 4.73i)14-s + (2.36 + 1.36i)15-s + (1.99 + 3.46i)16-s + (0.232 + 0.401i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.366 + 0.211i)3-s + (−0.866 − 0.5i)4-s + 1.66·5-s + (0.298 − 0.298i)6-s + (−1.13 + 0.654i)7-s + (−0.707 + 0.707i)8-s + (−0.410 − 0.711i)9-s + (0.431 − 1.61i)10-s + (0.301 − 0.522i)11-s + (−0.211 − 0.366i)12-s + (−0.720 + 0.693i)13-s + (0.338 + 1.26i)14-s + (0.610 + 0.352i)15-s + (0.499 + 0.866i)16-s + (0.0562 + 0.0974i)17-s + ⋯

Functional equation

Λ(s)=(104s/2 ΓC(s) L(s)=((0.494+0.869i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(104s/2 ΓC(s+1/2) L(s)=((0.494+0.869i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 104104    =    23⋅132^{3} \cdot 13
Sign: 0.494+0.869i0.494 + 0.869i
Analytic conductor: 0.8304440.830444
Root analytic conductor: 0.9112870.911287
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ104(69,⋅)\chi_{104} (69, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 104, ( :1/2), 0.494+0.869i)(2,\ 104,\ (\ :1/2),\ 0.494 + 0.869i)

Particular Values

L(1)L(1) ≈\approx 1.10059−0.640144i1.10059 - 0.640144i
L(12)L(\frac12) ≈\approx 1.10059−0.640144i1.10059 - 0.640144i
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+(2.59−2.5i)T 1 + (2.59 - 2.5i)T
good3 1+(−0.633−0.366i)T+(1.5+2.59i)T2 1 + (-0.633 - 0.366i)T + (1.5 + 2.59i)T^{2}
5 1−3.73T+5T2 1 - 3.73T + 5T^{2}
7 1+(3−1.73i)T+(3.5−6.06i)T2 1 + (3 - 1.73i)T + (3.5 - 6.06i)T^{2}
11 1+(−1+1.73i)T+(−5.5−9.52i)T2 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2}
17 1+(−0.232−0.401i)T+(−8.5+14.7i)T2 1 + (-0.232 - 0.401i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.633−1.09i)T+(−9.5+16.4i)T2 1 + (-0.633 - 1.09i)T + (-9.5 + 16.4i)T^{2}
23 1+(4.09−7.09i)T+(−11.5−19.9i)T2 1 + (4.09 - 7.09i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.59−1.5i)T+(14.5+25.1i)T2 1 + (-2.59 - 1.5i)T + (14.5 + 25.1i)T^{2}
31 1+4.73iT−31T2 1 + 4.73iT - 31T^{2}
37 1+(−2.13+3.69i)T+(−18.5−32.0i)T2 1 + (-2.13 + 3.69i)T + (-18.5 - 32.0i)T^{2}
41 1+(7.96+4.59i)T+(20.5+35.5i)T2 1 + (7.96 + 4.59i)T + (20.5 + 35.5i)T^{2}
43 1+(−2.19+1.26i)T+(21.5−37.2i)T2 1 + (-2.19 + 1.26i)T + (21.5 - 37.2i)T^{2}
47 1+6.73iT−47T2 1 + 6.73iT - 47T^{2}
53 1−3.92iT−53T2 1 - 3.92iT - 53T^{2}
59 1+(−0.267−0.464i)T+(−29.5+51.0i)T2 1 + (-0.267 - 0.464i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.866+0.5i)T+(30.5−52.8i)T2 1 + (-0.866 + 0.5i)T + (30.5 - 52.8i)T^{2}
67 1+(−3.63+6.29i)T+(−33.5−58.0i)T2 1 + (-3.63 + 6.29i)T + (-33.5 - 58.0i)T^{2}
71 1+(−8.02+4.63i)T+(35.5−61.4i)T2 1 + (-8.02 + 4.63i)T + (35.5 - 61.4i)T^{2}
73 1−1.73iT−73T2 1 - 1.73iT - 73T^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1−1.46T+83T2 1 - 1.46T + 83T^{2}
89 1+(−6.46−3.73i)T+(44.5+77.0i)T2 1 + (-6.46 - 3.73i)T + (44.5 + 77.0i)T^{2}
97 1+(5.19−3i)T+(48.5−84.0i)T2 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{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

−13.68487214625966252566828586197, −12.58792125577513754843320534867, −11.69272639658502696953682085000, −10.06997274265500595141798590178, −9.512756683316004384601837538598, −8.908559251459995551990744801681, −6.30479423701509434950843452068, −5.55034809440366847308401048876, −3.50044679182669776362616364966, −2.22246397514802082514205269828, 2.79563494400777199708628505245, 4.88613071800989277155250540128, 6.14559647065851798915674748456, 7.00963515152245142485097021304, 8.408860199936460454095466567285, 9.702696483346020277062097088550, 10.20821305767706023121690391563, 12.55157854028107888660076389757, 13.23259971866806694876623277241, 13.98495319258192499784782143244

Graph of the ZZ-function along the critical line