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Properties

Label 2-3960-1.1-c1-0-14
Degree 22
Conductor 39603960
Sign 11
Analytic cond. 31.620731.6207
Root an. cond. 5.623235.62323
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 1.41·7-s + 11-s − 0.585·13-s + 3.41·17-s − 5.65·19-s + 2.82·23-s + 25-s − 0.828·29-s + 6.48·31-s − 1.41·35-s − 7.65·37-s + 10.4·41-s + 2.58·43-s − 2.82·47-s − 5·49-s + 7.17·53-s − 55-s + 10.4·59-s − 3.17·61-s + 0.585·65-s − 8.48·67-s + 3.17·71-s + 7.41·73-s + 1.41·77-s + 1.65·79-s − 0.242·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.534·7-s + 0.301·11-s − 0.162·13-s + 0.828·17-s − 1.29·19-s + 0.589·23-s + 0.200·25-s − 0.153·29-s + 1.16·31-s − 0.239·35-s − 1.25·37-s + 1.63·41-s + 0.394·43-s − 0.412·47-s − 0.714·49-s + 0.985·53-s − 0.134·55-s + 1.36·59-s − 0.406·61-s + 0.0726·65-s − 1.03·67-s + 0.376·71-s + 0.867·73-s + 0.161·77-s + 0.186·79-s − 0.0266·83-s + ⋯

Functional equation

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

Invariants

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

Particular Values

L(1)L(1) ≈\approx 1.8321193341.832119334
L(12)L(\frac12) ≈\approx 1.8321193341.832119334
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+T 1 + T
11 1−T 1 - T
good7 1−1.41T+7T2 1 - 1.41T + 7T^{2}
13 1+0.585T+13T2 1 + 0.585T + 13T^{2}
17 1−3.41T+17T2 1 - 3.41T + 17T^{2}
19 1+5.65T+19T2 1 + 5.65T + 19T^{2}
23 1−2.82T+23T2 1 - 2.82T + 23T^{2}
29 1+0.828T+29T2 1 + 0.828T + 29T^{2}
31 1−6.48T+31T2 1 - 6.48T + 31T^{2}
37 1+7.65T+37T2 1 + 7.65T + 37T^{2}
41 1−10.4T+41T2 1 - 10.4T + 41T^{2}
43 1−2.58T+43T2 1 - 2.58T + 43T^{2}
47 1+2.82T+47T2 1 + 2.82T + 47T^{2}
53 1−7.17T+53T2 1 - 7.17T + 53T^{2}
59 1−10.4T+59T2 1 - 10.4T + 59T^{2}
61 1+3.17T+61T2 1 + 3.17T + 61T^{2}
67 1+8.48T+67T2 1 + 8.48T + 67T^{2}
71 1−3.17T+71T2 1 - 3.17T + 71T^{2}
73 1−7.41T+73T2 1 - 7.41T + 73T^{2}
79 1−1.65T+79T2 1 - 1.65T + 79T^{2}
83 1+0.242T+83T2 1 + 0.242T + 83T^{2}
89 1−4.34T+89T2 1 - 4.34T + 89T^{2}
97 1−0.828T+97T2 1 - 0.828T + 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

−8.410506837921309670529534113244, −7.79745959588708426682385365023, −7.04327952103669924828822244729, −6.30481694063817133506824655184, −5.42401850582041045434216929504, −4.61725189832187944956222852994, −3.94621828228751804059354460623, −2.98562958700781800534273445151, −1.95827998990836041620355231472, −0.790819635221559233414311353560, 0.790819635221559233414311353560, 1.95827998990836041620355231472, 2.98562958700781800534273445151, 3.94621828228751804059354460623, 4.61725189832187944956222852994, 5.42401850582041045434216929504, 6.30481694063817133506824655184, 7.04327952103669924828822244729, 7.79745959588708426682385365023, 8.410506837921309670529534113244

Graph of the ZZ-function along the critical line