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Properties

Label 2-4001-1.1-c1-0-128
Degree 22
Conductor 40014001
Sign 11
Analytic cond. 31.948131.9481
Root an. cond. 5.652265.65226
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  − 0.892·2-s − 0.475·3-s − 1.20·4-s + 4.01·5-s + 0.424·6-s − 0.567·7-s + 2.85·8-s − 2.77·9-s − 3.58·10-s + 4.71·11-s + 0.571·12-s + 0.480·13-s + 0.506·14-s − 1.90·15-s − 0.146·16-s + 3.16·17-s + 2.47·18-s + 4.33·19-s − 4.82·20-s + 0.269·21-s − 4.20·22-s − 6.76·23-s − 1.35·24-s + 11.1·25-s − 0.429·26-s + 2.74·27-s + 0.683·28-s + ⋯
L(s)  = 1  − 0.631·2-s − 0.274·3-s − 0.601·4-s + 1.79·5-s + 0.173·6-s − 0.214·7-s + 1.01·8-s − 0.924·9-s − 1.13·10-s + 1.42·11-s + 0.164·12-s + 0.133·13-s + 0.135·14-s − 0.492·15-s − 0.0366·16-s + 0.767·17-s + 0.583·18-s + 0.994·19-s − 1.07·20-s + 0.0588·21-s − 0.897·22-s − 1.41·23-s − 0.277·24-s + 2.22·25-s − 0.0841·26-s + 0.527·27-s + 0.129·28-s + ⋯

Functional equation

Λ(s)=(4001s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4001s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40014001
Sign: 11
Analytic conductor: 31.948131.9481
Root analytic conductor: 5.652265.65226
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4001, ( :1/2), 1)(2,\ 4001,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6285173111.628517311
L(12)L(\frac12) ≈\approx 1.6285173111.628517311
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)
bad4001 1+O(T) 1+O(T)
good2 1+0.892T+2T2 1 + 0.892T + 2T^{2}
3 1+0.475T+3T2 1 + 0.475T + 3T^{2}
5 1−4.01T+5T2 1 - 4.01T + 5T^{2}
7 1+0.567T+7T2 1 + 0.567T + 7T^{2}
11 1−4.71T+11T2 1 - 4.71T + 11T^{2}
13 1−0.480T+13T2 1 - 0.480T + 13T^{2}
17 1−3.16T+17T2 1 - 3.16T + 17T^{2}
19 1−4.33T+19T2 1 - 4.33T + 19T^{2}
23 1+6.76T+23T2 1 + 6.76T + 23T^{2}
29 1−8.94T+29T2 1 - 8.94T + 29T^{2}
31 1−5.48T+31T2 1 - 5.48T + 31T^{2}
37 1+8.35T+37T2 1 + 8.35T + 37T^{2}
41 1−4.43T+41T2 1 - 4.43T + 41T^{2}
43 1−0.890T+43T2 1 - 0.890T + 43T^{2}
47 1+5.98T+47T2 1 + 5.98T + 47T^{2}
53 1+3.04T+53T2 1 + 3.04T + 53T^{2}
59 1−14.5T+59T2 1 - 14.5T + 59T^{2}
61 1−0.262T+61T2 1 - 0.262T + 61T^{2}
67 1+9.92T+67T2 1 + 9.92T + 67T^{2}
71 1−14.4T+71T2 1 - 14.4T + 71T^{2}
73 1−6.46T+73T2 1 - 6.46T + 73T^{2}
79 1+6.44T+79T2 1 + 6.44T + 79T^{2}
83 1−13.8T+83T2 1 - 13.8T + 83T^{2}
89 1−6.07T+89T2 1 - 6.07T + 89T^{2}
97 1+9.15T+97T2 1 + 9.15T + 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

−8.570315706786230204763785649959, −8.040103489708942037553867019680, −6.71107378256092318480917154091, −6.31582026540283134708967845712, −5.50284793320222838141876405299, −4.96969383370784598533531362607, −3.81635880675668823828951283942, −2.80537046324061186864941574385, −1.64221486042685238468882208153, −0.898501735722457882071198475256, 0.898501735722457882071198475256, 1.64221486042685238468882208153, 2.80537046324061186864941574385, 3.81635880675668823828951283942, 4.96969383370784598533531362607, 5.50284793320222838141876405299, 6.31582026540283134708967845712, 6.71107378256092318480917154091, 8.040103489708942037553867019680, 8.570315706786230204763785649959

Graph of the ZZ-function along the critical line