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Properties

Label 2-4001-1.1-c1-0-8
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  − 2.61·2-s − 0.259·3-s + 4.81·4-s − 4.09·5-s + 0.678·6-s + 2.08·7-s − 7.34·8-s − 2.93·9-s + 10.6·10-s − 4.54·11-s − 1.25·12-s − 2.98·13-s − 5.45·14-s + 1.06·15-s + 9.55·16-s + 4.39·17-s + 7.65·18-s − 0.751·19-s − 19.7·20-s − 0.542·21-s + 11.8·22-s − 0.781·23-s + 1.90·24-s + 11.7·25-s + 7.78·26-s + 1.54·27-s + 10.0·28-s + ⋯
L(s)  = 1  − 1.84·2-s − 0.149·3-s + 2.40·4-s − 1.83·5-s + 0.276·6-s + 0.789·7-s − 2.59·8-s − 0.977·9-s + 3.37·10-s − 1.37·11-s − 0.361·12-s − 0.826·13-s − 1.45·14-s + 0.274·15-s + 2.38·16-s + 1.06·17-s + 1.80·18-s − 0.172·19-s − 4.40·20-s − 0.118·21-s + 2.52·22-s − 0.162·23-s + 0.389·24-s + 2.35·25-s + 1.52·26-s + 0.296·27-s + 1.89·28-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 0.023173141730.02317314173
L(12)L(\frac12) ≈\approx 0.023173141730.02317314173
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+2.61T+2T2 1 + 2.61T + 2T^{2}
3 1+0.259T+3T2 1 + 0.259T + 3T^{2}
5 1+4.09T+5T2 1 + 4.09T + 5T^{2}
7 1−2.08T+7T2 1 - 2.08T + 7T^{2}
11 1+4.54T+11T2 1 + 4.54T + 11T^{2}
13 1+2.98T+13T2 1 + 2.98T + 13T^{2}
17 1−4.39T+17T2 1 - 4.39T + 17T^{2}
19 1+0.751T+19T2 1 + 0.751T + 19T^{2}
23 1+0.781T+23T2 1 + 0.781T + 23T^{2}
29 1+4.78T+29T2 1 + 4.78T + 29T^{2}
31 1+3.90T+31T2 1 + 3.90T + 31T^{2}
37 1+6.77T+37T2 1 + 6.77T + 37T^{2}
41 1+11.6T+41T2 1 + 11.6T + 41T^{2}
43 1−0.552T+43T2 1 - 0.552T + 43T^{2}
47 1+1.35T+47T2 1 + 1.35T + 47T^{2}
53 1+8.71T+53T2 1 + 8.71T + 53T^{2}
59 1+11.9T+59T2 1 + 11.9T + 59T^{2}
61 1+0.470T+61T2 1 + 0.470T + 61T^{2}
67 1+8.34T+67T2 1 + 8.34T + 67T^{2}
71 1−10.0T+71T2 1 - 10.0T + 71T^{2}
73 1−7.49T+73T2 1 - 7.49T + 73T^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1+13.4T+83T2 1 + 13.4T + 83T^{2}
89 1−11.9T+89T2 1 - 11.9T + 89T^{2}
97 1+18.5T+97T2 1 + 18.5T + 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

−8.268326236063130669308822971407, −7.85171910890825771439941783451, −7.53998433678162739301622800888, −6.75292872494400020860570808565, −5.49802359397779796042157286942, −4.85860068794577940660273861650, −3.45012709302007568987307312375, −2.79291456173214772248973211945, −1.61696308614142262885175431345, −0.11307431033662277385962161195, 0.11307431033662277385962161195, 1.61696308614142262885175431345, 2.79291456173214772248973211945, 3.45012709302007568987307312375, 4.85860068794577940660273861650, 5.49802359397779796042157286942, 6.75292872494400020860570808565, 7.53998433678162739301622800888, 7.85171910890825771439941783451, 8.268326236063130669308822971407

Graph of the ZZ-function along the critical line