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Properties

Label 2-1560-13.3-c1-0-26
Degree 22
Conductor 15601560
Sign −0.994−0.106i-0.994 - 0.106i
Analytic cond. 12.456612.4566
Root an. cond. 3.529393.52939
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  + (−0.5 − 0.866i)3-s − 5-s + (0.155 − 0.269i)7-s + (−0.499 + 0.866i)9-s + (−2.21 − 3.83i)11-s + (2.82 − 2.24i)13-s + (0.5 + 0.866i)15-s + (0.344 − 0.596i)17-s + (−1.76 + 3.05i)19-s − 0.311·21-s + (0.622 + 1.07i)23-s + 25-s + 0.999·27-s + (−0.418 − 0.724i)29-s − 6.33·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s − 0.447·5-s + (0.0587 − 0.101i)7-s + (−0.166 + 0.288i)9-s + (−0.667 − 1.15i)11-s + (0.782 − 0.622i)13-s + (0.129 + 0.223i)15-s + (0.0835 − 0.144i)17-s + (−0.404 + 0.700i)19-s − 0.0678·21-s + (0.129 + 0.224i)23-s + 0.200·25-s + 0.192·27-s + (−0.0776 − 0.134i)29-s − 1.13·31-s + ⋯

Functional equation

Λ(s)=(1560s/2 ΓC(s) L(s)=((−0.994−0.106i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1560s/2 ΓC(s+1/2) L(s)=((−0.994−0.106i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15601560    =    23⋅3⋅5⋅132^{3} \cdot 3 \cdot 5 \cdot 13
Sign: −0.994−0.106i-0.994 - 0.106i
Analytic conductor: 12.456612.4566
Root analytic conductor: 3.529393.52939
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1560(601,⋅)\chi_{1560} (601, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1560, ( :1/2), −0.994−0.106i)(2,\ 1560,\ (\ :1/2),\ -0.994 - 0.106i)

Particular Values

L(1)L(1) ≈\approx 0.40799383110.4079938311
L(12)L(\frac12) ≈\approx 0.40799383110.4079938311
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
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
5 1+T 1 + T
13 1+(−2.82+2.24i)T 1 + (-2.82 + 2.24i)T
good7 1+(−0.155+0.269i)T+(−3.5−6.06i)T2 1 + (-0.155 + 0.269i)T + (-3.5 - 6.06i)T^{2}
11 1+(2.21+3.83i)T+(−5.5+9.52i)T2 1 + (2.21 + 3.83i)T + (-5.5 + 9.52i)T^{2}
17 1+(−0.344+0.596i)T+(−8.5−14.7i)T2 1 + (-0.344 + 0.596i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.76−3.05i)T+(−9.5−16.4i)T2 1 + (1.76 - 3.05i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.622−1.07i)T+(−11.5+19.9i)T2 1 + (-0.622 - 1.07i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.418+0.724i)T+(−14.5+25.1i)T2 1 + (0.418 + 0.724i)T + (-14.5 + 25.1i)T^{2}
31 1+6.33T+31T2 1 + 6.33T + 31T^{2}
37 1+(−1.90−3.29i)T+(−18.5+32.0i)T2 1 + (-1.90 - 3.29i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.48+6.03i)T+(−20.5+35.5i)T2 1 + (3.48 + 6.03i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.821−1.42i)T+(−21.5−37.2i)T2 1 + (0.821 - 1.42i)T + (-21.5 - 37.2i)T^{2}
47 1+12.5T+47T2 1 + 12.5T + 47T^{2}
53 1+10.2T+53T2 1 + 10.2T + 53T^{2}
59 1+(−5.38+9.33i)T+(−29.5−51.0i)T2 1 + (-5.38 + 9.33i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.57+2.72i)T+(−30.5−52.8i)T2 1 + (-1.57 + 2.72i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.98+6.90i)T+(−33.5+58.0i)T2 1 + (3.98 + 6.90i)T + (-33.5 + 58.0i)T^{2}
71 1+(1.35−2.34i)T+(−35.5−61.4i)T2 1 + (1.35 - 2.34i)T + (-35.5 - 61.4i)T^{2}
73 1+14.6T+73T2 1 + 14.6T + 73T^{2}
79 1+9.28T+79T2 1 + 9.28T + 79T^{2}
83 1+1.61T+83T2 1 + 1.61T + 83T^{2}
89 1+(−4.15−7.20i)T+(−44.5+77.0i)T2 1 + (-4.15 - 7.20i)T + (-44.5 + 77.0i)T^{2}
97 1+(3.51−6.08i)T+(−48.5−84.0i)T2 1 + (3.51 - 6.08i)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

−8.768538488559874327900716837404, −8.134320783144465616672089411165, −7.59823492571704688082723328065, −6.50679454036682773534648300018, −5.80638757480746913760195994723, −5.03331338889914562035381242939, −3.73978572949254583236012041143, −2.99204067548909184174018656544, −1.50837681931920008591702087521, −0.16530900912215639196768494277, 1.73096830628546984535007835548, 3.01724763383617153557560617048, 4.13410662274115770094500859571, 4.74162344804541908458133363049, 5.66812117770916360999145150789, 6.69762252698996496196790402398, 7.37199868212369332303875929321, 8.373731762820050213430298777826, 9.052199735399696364783354282042, 9.902972988844415168245109970523

Graph of the ZZ-function along the critical line