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Properties

Label 2-4002-1.1-c1-0-43
Degree 22
Conductor 40024002
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 2.25·5-s + 6-s + 1.35·7-s − 8-s + 9-s + 2.25·10-s + 1.14·11-s − 12-s + 0.110·13-s − 1.35·14-s + 2.25·15-s + 16-s + 7.57·17-s − 18-s − 7.57·19-s − 2.25·20-s − 1.35·21-s − 1.14·22-s − 23-s + 24-s + 0.0989·25-s − 0.110·26-s − 27-s + 1.35·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.00·5-s + 0.408·6-s + 0.512·7-s − 0.353·8-s + 0.333·9-s + 0.714·10-s + 0.346·11-s − 0.288·12-s + 0.0306·13-s − 0.362·14-s + 0.583·15-s + 0.250·16-s + 1.83·17-s − 0.235·18-s − 1.73·19-s − 0.504·20-s − 0.295·21-s − 0.244·22-s − 0.208·23-s + 0.204·24-s + 0.0197·25-s − 0.0216·26-s − 0.192·27-s + 0.256·28-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: −1-1
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: 11
Selberg data: (2, 4002, ( :1/2), −1)(2,\ 4002,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+2.25T+5T2 1 + 2.25T + 5T^{2}
7 1−1.35T+7T2 1 - 1.35T + 7T^{2}
11 1−1.14T+11T2 1 - 1.14T + 11T^{2}
13 1−0.110T+13T2 1 - 0.110T + 13T^{2}
17 1−7.57T+17T2 1 - 7.57T + 17T^{2}
19 1+7.57T+19T2 1 + 7.57T + 19T^{2}
31 1+10.2T+31T2 1 + 10.2T + 31T^{2}
37 1−11.0T+37T2 1 - 11.0T + 37T^{2}
41 1+6.08T+41T2 1 + 6.08T + 41T^{2}
43 1+7.97T+43T2 1 + 7.97T + 43T^{2}
47 1−4.67T+47T2 1 - 4.67T + 47T^{2}
53 1−10.7T+53T2 1 - 10.7T + 53T^{2}
59 1−11.8T+59T2 1 - 11.8T + 59T^{2}
61 1−14.6T+61T2 1 - 14.6T + 61T^{2}
67 1+5.32T+67T2 1 + 5.32T + 67T^{2}
71 1+1.59T+71T2 1 + 1.59T + 71T^{2}
73 1−10.5T+73T2 1 - 10.5T + 73T^{2}
79 1−0.564T+79T2 1 - 0.564T + 79T^{2}
83 1+12.2T+83T2 1 + 12.2T + 83T^{2}
89 1+0.540T+89T2 1 + 0.540T + 89T^{2}
97 1+14.7T+97T2 1 + 14.7T + 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

−8.171874366711058341682786789228, −7.42912316872687614618812014257, −6.83273811015277338609180970804, −5.88859825357200398995746609355, −5.21957185567357356586545133267, −4.09950670220688562053097865653, −3.60961268161993976456194889568, −2.21989771787079044870588828463, −1.15985899128314676835907903095, 0, 1.15985899128314676835907903095, 2.21989771787079044870588828463, 3.60961268161993976456194889568, 4.09950670220688562053097865653, 5.21957185567357356586545133267, 5.88859825357200398995746609355, 6.83273811015277338609180970804, 7.42912316872687614618812014257, 8.171874366711058341682786789228

Graph of the ZZ-function along the critical line