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Properties

Label 2-104-104.69-c1-0-9
Degree 22
Conductor 104104
Sign −0.869+0.494i-0.869 + 0.494i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−0.633 − 0.366i)3-s − 2i·4-s − 3.73·5-s + (1 − 0.267i)6-s + (−3 + 1.73i)7-s + (2 + 2i)8-s + (−1.23 − 2.13i)9-s + (3.73 − 3.73i)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 − 4·16-s + (0.232 + 0.401i)17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.366 − 0.211i)3-s − i·4-s − 1.66·5-s + (0.408 − 0.109i)6-s + (−1.13 + 0.654i)7-s + (0.707 + 0.707i)8-s + (−0.410 − 0.711i)9-s + (1.18 − 1.18i)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 − 16-s + (0.0562 + 0.0974i)17-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 104104    =    23⋅132^{3} \cdot 13
Sign: −0.869+0.494i-0.869 + 0.494i
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: 11
Selberg data: (2, 104, ( :1/2), −0.869+0.494i)(2,\ 104,\ (\ :1/2),\ -0.869 + 0.494i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+(−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

−13.27459514992018053723281735713, −12.09155432331919422560021806903, −11.34658401651644295602498313189, −9.959275773227621639853646702370, −8.786923276720999196671280305504, −7.79482891601130147892503056776, −6.70241883468960821824616526454, −5.57173060992550828612275010275, −3.58033679028000759983194361425, 0, 3.28122465418330377340162477398, 4.31439426193152263666412656159, 6.67002848047013952318864383646, 7.890101517039449672594169451293, 8.740548967189365823097484865780, 10.32069971646691621635120490554, 10.99008065912122407362118649739, 11.87422779914322198786002628128, 12.82843358379086968609241695444

Graph of the ZZ-function along the critical line