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Properties

Label 2-4002-1.1-c1-0-38
Degree 22
Conductor 40024002
Sign 11
Analytic cond. 31.956131.9561
Root an. cond. 5.652975.65297
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-s − 3-s + 4-s − 0.869·5-s − 6-s + 3.97·7-s + 8-s + 9-s − 0.869·10-s + 3.41·11-s − 12-s + 3.88·13-s + 3.97·14-s + 0.869·15-s + 16-s − 1.10·17-s + 18-s + 3.97·19-s − 0.869·20-s − 3.97·21-s + 3.41·22-s + 23-s − 24-s − 4.24·25-s + 3.88·26-s − 27-s + 3.97·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.388·5-s − 0.408·6-s + 1.50·7-s + 0.353·8-s + 0.333·9-s − 0.275·10-s + 1.02·11-s − 0.288·12-s + 1.07·13-s + 1.06·14-s + 0.224·15-s + 0.250·16-s − 0.267·17-s + 0.235·18-s + 0.911·19-s − 0.194·20-s − 0.866·21-s + 0.727·22-s + 0.208·23-s − 0.204·24-s − 0.848·25-s + 0.762·26-s − 0.192·27-s + 0.750·28-s + ⋯

Functional equation

Λ(s)=(4002s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4002s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40024002    =    2⋅3⋅23⋅292 \cdot 3 \cdot 23 \cdot 29
Sign: 11
Analytic conductor: 31.956131.9561
Root analytic conductor: 5.652975.65297
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4002, ( :1/2), 1)(2,\ 4002,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3493341363.349334136
L(12)L(\frac12) ≈\approx 3.3493341363.349334136
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−T 1 - T
3 1+T 1 + T
23 1−T 1 - T
29 1−T 1 - T
good5 1+0.869T+5T2 1 + 0.869T + 5T^{2}
7 1−3.97T+7T2 1 - 3.97T + 7T^{2}
11 1−3.41T+11T2 1 - 3.41T + 11T^{2}
13 1−3.88T+13T2 1 - 3.88T + 13T^{2}
17 1+1.10T+17T2 1 + 1.10T + 17T^{2}
19 1−3.97T+19T2 1 - 3.97T + 19T^{2}
31 1−10.7T+31T2 1 - 10.7T + 31T^{2}
37 1+10.8T+37T2 1 + 10.8T + 37T^{2}
41 1+12.2T+41T2 1 + 12.2T + 41T^{2}
43 1−8.48T+43T2 1 - 8.48T + 43T^{2}
47 1−6.30T+47T2 1 - 6.30T + 47T^{2}
53 1−3.72T+53T2 1 - 3.72T + 53T^{2}
59 1+9.89T+59T2 1 + 9.89T + 59T^{2}
61 1−8.38T+61T2 1 - 8.38T + 61T^{2}
67 1+11.7T+67T2 1 + 11.7T + 67T^{2}
71 1+3.08T+71T2 1 + 3.08T + 71T^{2}
73 1+14.2T+73T2 1 + 14.2T + 73T^{2}
79 1−0.168T+79T2 1 - 0.168T + 79T^{2}
83 1+7.07T+83T2 1 + 7.07T + 83T^{2}
89 1−6.49T+89T2 1 - 6.49T + 89T^{2}
97 1+0.491T+97T2 1 + 0.491T + 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

−8.406758408923459893120128221948, −7.57973190210378481679420900321, −6.90688716711925929392255360561, −6.09537798794970417396990177968, −5.42324909570342058504740865697, −4.60262574261817478154861361486, −4.10215985565995752891967226197, −3.19972358635168019337874589509, −1.78599504935920333233772536024, −1.09839432668947993786859312568, 1.09839432668947993786859312568, 1.78599504935920333233772536024, 3.19972358635168019337874589509, 4.10215985565995752891967226197, 4.60262574261817478154861361486, 5.42324909570342058504740865697, 6.09537798794970417396990177968, 6.90688716711925929392255360561, 7.57973190210378481679420900321, 8.406758408923459893120128221948

Graph of the ZZ-function along the critical line