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Properties

Label 2-104-13.3-c1-0-2
Degree 22
Conductor 104104
Sign −0.597+0.802i-0.597 + 0.802i
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  + (−1.28 − 2.21i)3-s − 3.56·5-s + (1.28 − 2.21i)7-s + (−1.78 + 3.08i)9-s + (−1.28 − 2.21i)11-s + (3.34 + 1.35i)13-s + (4.56 + 7.90i)15-s + (2.5 − 4.33i)17-s + (1.28 − 2.21i)19-s − 6.56·21-s + (1.84 + 3.19i)23-s + 7.68·25-s + 1.43·27-s + (2.5 + 4.33i)29-s − 8·31-s + ⋯
L(s)  = 1  + (−0.739 − 1.28i)3-s − 1.59·5-s + (0.484 − 0.838i)7-s + (−0.593 + 1.02i)9-s + (−0.386 − 0.668i)11-s + (0.926 + 0.375i)13-s + (1.17 + 2.03i)15-s + (0.606 − 1.05i)17-s + (0.293 − 0.508i)19-s − 1.43·21-s + (0.384 + 0.665i)23-s + 1.53·25-s + 0.276·27-s + (0.464 + 0.804i)29-s − 1.43·31-s + ⋯

Functional equation

Λ(s)=(104s/2 ΓC(s) L(s)=((−0.597+0.802i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(104s/2 ΓC(s+1/2) L(s)=((−0.597+0.802i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 104104    =    23⋅132^{3} \cdot 13
Sign: −0.597+0.802i-0.597 + 0.802i
Analytic conductor: 0.8304440.830444
Root analytic conductor: 0.9112870.911287
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ104(81,⋅)\chi_{104} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 104, ( :1/2), −0.597+0.802i)(2,\ 104,\ (\ :1/2),\ -0.597 + 0.802i)

Particular Values

L(1)L(1) ≈\approx 0.273861−0.545241i0.273861 - 0.545241i
L(12)L(\frac12) ≈\approx 0.273861−0.545241i0.273861 - 0.545241i
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
13 1+(−3.34−1.35i)T 1 + (-3.34 - 1.35i)T
good3 1+(1.28+2.21i)T+(−1.5+2.59i)T2 1 + (1.28 + 2.21i)T + (-1.5 + 2.59i)T^{2}
5 1+3.56T+5T2 1 + 3.56T + 5T^{2}
7 1+(−1.28+2.21i)T+(−3.5−6.06i)T2 1 + (-1.28 + 2.21i)T + (-3.5 - 6.06i)T^{2}
11 1+(1.28+2.21i)T+(−5.5+9.52i)T2 1 + (1.28 + 2.21i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2.5+4.33i)T+(−8.5−14.7i)T2 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.28+2.21i)T+(−9.5−16.4i)T2 1 + (-1.28 + 2.21i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.84−3.19i)T+(−11.5+19.9i)T2 1 + (-1.84 - 3.19i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.5−4.33i)T+(−14.5+25.1i)T2 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2}
31 1+8T+31T2 1 + 8T + 31T^{2}
37 1+(−0.5−0.866i)T+(−18.5+32.0i)T2 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2}
41 1+(4.62+8.00i)T+(−20.5+35.5i)T2 1 + (4.62 + 8.00i)T + (-20.5 + 35.5i)T^{2}
43 1+(−3.28+5.68i)T+(−21.5−37.2i)T2 1 + (-3.28 + 5.68i)T + (-21.5 - 37.2i)T^{2}
47 1−4T+47T2 1 - 4T + 47T^{2}
53 1−4.43T+53T2 1 - 4.43T + 53T^{2}
59 1+(−1.28+2.21i)T+(−29.5−51.0i)T2 1 + (-1.28 + 2.21i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.62+6.27i)T+(−30.5−52.8i)T2 1 + (-3.62 + 6.27i)T + (-30.5 - 52.8i)T^{2}
67 1+(−4.71−8.17i)T+(−33.5+58.0i)T2 1 + (-4.71 - 8.17i)T + (-33.5 + 58.0i)T^{2}
71 1+(3.84−6.65i)T+(−35.5−61.4i)T2 1 + (3.84 - 6.65i)T + (-35.5 - 61.4i)T^{2}
73 1+1.31T+73T2 1 + 1.31T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1−2.24T+83T2 1 - 2.24T + 83T^{2}
89 1+(−4.84−8.38i)T+(−44.5+77.0i)T2 1 + (-4.84 - 8.38i)T + (-44.5 + 77.0i)T^{2}
97 1+(1.40−2.43i)T+(−48.5−84.0i)T2 1 + (1.40 - 2.43i)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

−13.32256368666083898785419265553, −12.20264343955088622458999806498, −11.40427778897318641134569004509, −10.88451733391917141342322231506, −8.649601413668202888360217879250, −7.47728287435516302733903595920, −7.07870713237269886047931264198, −5.34556012197174859039546041200, −3.67044517799898412177113530074, −0.824650638140039589805644693019, 3.62620327595018432657970314984, 4.65923226360846828249036996429, 5.84348520248895630808699798970, 7.75463579543731121146012664390, 8.695766312726768034377928823489, 10.17731662745849353319352739485, 11.05986231834894520949000910039, 11.81157797143478261316024493523, 12.71389017682631377227128545157, 14.85003394814230466269313400817

Graph of the ZZ-function along the critical line