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Properties

Label 2-4001-1.1-c1-0-224
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 + 2.84·3-s + 4.83·4-s + 0.542·5-s − 7.44·6-s + 4.00·7-s − 7.41·8-s + 5.11·9-s − 1.41·10-s + 4.26·11-s + 13.7·12-s + 5.82·13-s − 10.4·14-s + 1.54·15-s + 9.71·16-s − 4.23·17-s − 13.3·18-s + 6.94·19-s + 2.62·20-s + 11.4·21-s − 11.1·22-s − 5.99·23-s − 21.1·24-s − 4.70·25-s − 15.2·26-s + 6.01·27-s + 19.3·28-s + ⋯
L(s)  = 1  − 1.84·2-s + 1.64·3-s + 2.41·4-s + 0.242·5-s − 3.04·6-s + 1.51·7-s − 2.62·8-s + 1.70·9-s − 0.448·10-s + 1.28·11-s + 3.97·12-s + 1.61·13-s − 2.79·14-s + 0.398·15-s + 2.42·16-s − 1.02·17-s − 3.15·18-s + 1.59·19-s + 0.586·20-s + 2.48·21-s − 2.37·22-s − 1.25·23-s − 4.31·24-s − 0.941·25-s − 2.98·26-s + 1.15·27-s + 3.66·28-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 2.3578579852.357857985
L(12)L(\frac12) ≈\approx 2.3578579852.357857985
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 1−2.84T+3T2 1 - 2.84T + 3T^{2}
5 1−0.542T+5T2 1 - 0.542T + 5T^{2}
7 1−4.00T+7T2 1 - 4.00T + 7T^{2}
11 1−4.26T+11T2 1 - 4.26T + 11T^{2}
13 1−5.82T+13T2 1 - 5.82T + 13T^{2}
17 1+4.23T+17T2 1 + 4.23T + 17T^{2}
19 1−6.94T+19T2 1 - 6.94T + 19T^{2}
23 1+5.99T+23T2 1 + 5.99T + 23T^{2}
29 1+3.54T+29T2 1 + 3.54T + 29T^{2}
31 1−9.01T+31T2 1 - 9.01T + 31T^{2}
37 1+5.38T+37T2 1 + 5.38T + 37T^{2}
41 1−5.31T+41T2 1 - 5.31T + 41T^{2}
43 1−2.25T+43T2 1 - 2.25T + 43T^{2}
47 1−3.58T+47T2 1 - 3.58T + 47T^{2}
53 1+13.1T+53T2 1 + 13.1T + 53T^{2}
59 1+7.82T+59T2 1 + 7.82T + 59T^{2}
61 1−1.46T+61T2 1 - 1.46T + 61T^{2}
67 1+4.09T+67T2 1 + 4.09T + 67T^{2}
71 1−13.6T+71T2 1 - 13.6T + 71T^{2}
73 1+9.91T+73T2 1 + 9.91T + 73T^{2}
79 1+4.77T+79T2 1 + 4.77T + 79T^{2}
83 1+12.5T+83T2 1 + 12.5T + 83T^{2}
89 1−17.5T+89T2 1 - 17.5T + 89T^{2}
97 1+11.9T+97T2 1 + 11.9T + 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

−8.486660183358852980999576906244, −7.978398572752588591225072055759, −7.58368016532377087964816709308, −6.63480236728174877918278112908, −5.88751309773272995410400757885, −4.33183739836957037295187293740, −3.56207373707181882319408347555, −2.48149903705243565732996765624, −1.57954188456051847488092121893, −1.33394055734303340366189478329, 1.33394055734303340366189478329, 1.57954188456051847488092121893, 2.48149903705243565732996765624, 3.56207373707181882319408347555, 4.33183739836957037295187293740, 5.88751309773272995410400757885, 6.63480236728174877918278112908, 7.58368016532377087964816709308, 7.978398572752588591225072055759, 8.486660183358852980999576906244

Graph of the ZZ-function along the critical line