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Properties

Label 2-3960-5.4-c1-0-67
Degree 22
Conductor 39603960
Sign −0.998+0.0452i-0.998 + 0.0452i
Analytic cond. 31.620731.6207
Root an. cond. 5.623235.62323
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  + (−0.101 − 2.23i)5-s − 3.61i·7-s − 11-s − 0.816i·13-s − 1.81i·17-s − 0.723·19-s − 4.63i·23-s + (−4.97 + 0.451i)25-s + 0.539·29-s + 9.04·31-s + (−8.07 + 0.365i)35-s − 9.34i·37-s − 1.11·41-s + 5.63i·43-s − 2.01i·47-s + ⋯
L(s)  = 1  + (−0.0452 − 0.998i)5-s − 1.36i·7-s − 0.301·11-s − 0.226i·13-s − 0.440i·17-s − 0.165·19-s − 0.966i·23-s + (−0.995 + 0.0903i)25-s + 0.100·29-s + 1.62·31-s + (−1.36 + 0.0617i)35-s − 1.53i·37-s − 0.173·41-s + 0.859i·43-s − 0.294i·47-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((−0.998+0.0452i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0452i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s+1/2) L(s)=((−0.998+0.0452i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0452i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: −0.998+0.0452i-0.998 + 0.0452i
Analytic conductor: 31.620731.6207
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3960(3169,⋅)\chi_{3960} (3169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :1/2), −0.998+0.0452i)(2,\ 3960,\ (\ :1/2),\ -0.998 + 0.0452i)

Particular Values

L(1)L(1) ≈\approx 1.2137636511.213763651
L(12)L(\frac12) ≈\approx 1.2137636511.213763651
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 1
5 1+(0.101+2.23i)T 1 + (0.101 + 2.23i)T
11 1+T 1 + T
good7 1+3.61iT−7T2 1 + 3.61iT - 7T^{2}
13 1+0.816iT−13T2 1 + 0.816iT - 13T^{2}
17 1+1.81iT−17T2 1 + 1.81iT - 17T^{2}
19 1+0.723T+19T2 1 + 0.723T + 19T^{2}
23 1+4.63iT−23T2 1 + 4.63iT - 23T^{2}
29 1−0.539T+29T2 1 - 0.539T + 29T^{2}
31 1−9.04T+31T2 1 - 9.04T + 31T^{2}
37 1+9.34iT−37T2 1 + 9.34iT - 37T^{2}
41 1+1.11T+41T2 1 + 1.11T + 41T^{2}
43 1−5.63iT−43T2 1 - 5.63iT - 43T^{2}
47 1+2.01iT−47T2 1 + 2.01iT - 47T^{2}
53 1+1.29iT−53T2 1 + 1.29iT - 53T^{2}
59 1+7.11T+59T2 1 + 7.11T + 59T^{2}
61 1−3.61T+61T2 1 - 3.61T + 61T^{2}
67 1−2.26iT−67T2 1 - 2.26iT - 67T^{2}
71 1+11.8T+71T2 1 + 11.8T + 71T^{2}
73 1+11.3iT−73T2 1 + 11.3iT - 73T^{2}
79 1−3.97T+79T2 1 - 3.97T + 79T^{2}
83 1−2.50iT−83T2 1 - 2.50iT - 83T^{2}
89 1+6.19T+89T2 1 + 6.19T + 89T^{2}
97 1−0.0431iT−97T2 1 - 0.0431iT - 97T^{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

−8.028852081979971431458356147104, −7.51715643929440416266875472542, −6.67245136088507889008406678813, −5.86867735404551697887436885753, −4.83792775144416372279798957657, −4.44098523389621416629952339175, −3.58749476333353050667874754032, −2.45585132180528955356551279209, −1.17624228403865209930388803518, −0.36979971531639552493704920819, 1.64237019882859400662756937905, 2.61716776966925762533234054920, 3.15162807953014815920162333063, 4.24031132956456808128609004035, 5.21907877624769766392970568079, 5.97033151520452688486154517765, 6.49998633559746209478241185339, 7.33710350469875058697906346562, 8.178029604053932554450966592555, 8.685870821520992763738211509154

Graph of the ZZ-function along the critical line