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Properties

Label 2-4002-1.1-c1-0-27
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 − 2.88·5-s + 6-s − 3.71·7-s − 8-s + 9-s + 2.88·10-s + 5.50·11-s − 12-s − 3.61·13-s + 3.71·14-s + 2.88·15-s + 16-s − 4.95·17-s − 18-s + 4.95·19-s − 2.88·20-s + 3.71·21-s − 5.50·22-s − 23-s + 24-s + 3.33·25-s + 3.61·26-s − 27-s − 3.71·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.29·5-s + 0.408·6-s − 1.40·7-s − 0.353·8-s + 0.333·9-s + 0.913·10-s + 1.66·11-s − 0.288·12-s − 1.00·13-s + 0.993·14-s + 0.745·15-s + 0.250·16-s − 1.20·17-s − 0.235·18-s + 1.13·19-s − 0.645·20-s + 0.810·21-s − 1.17·22-s − 0.208·23-s + 0.204·24-s + 0.667·25-s + 0.709·26-s − 0.192·27-s − 0.702·28-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+2.88T+5T2 1 + 2.88T + 5T^{2}
7 1+3.71T+7T2 1 + 3.71T + 7T^{2}
11 1−5.50T+11T2 1 - 5.50T + 11T^{2}
13 1+3.61T+13T2 1 + 3.61T + 13T^{2}
17 1+4.95T+17T2 1 + 4.95T + 17T^{2}
19 1−4.95T+19T2 1 - 4.95T + 19T^{2}
31 1−6.02T+31T2 1 - 6.02T + 31T^{2}
37 1+1.34T+37T2 1 + 1.34T + 37T^{2}
41 1+1.77T+41T2 1 + 1.77T + 41T^{2}
43 1+7.39T+43T2 1 + 7.39T + 43T^{2}
47 1−13.6T+47T2 1 - 13.6T + 47T^{2}
53 1−2.70T+53T2 1 - 2.70T + 53T^{2}
59 1−8.87T+59T2 1 - 8.87T + 59T^{2}
61 1+5.99T+61T2 1 + 5.99T + 61T^{2}
67 1−9.67T+67T2 1 - 9.67T + 67T^{2}
71 1−3.39T+71T2 1 - 3.39T + 71T^{2}
73 1−13.9T+73T2 1 - 13.9T + 73T^{2}
79 1+4.99T+79T2 1 + 4.99T + 79T^{2}
83 1−9.98T+83T2 1 - 9.98T + 83T^{2}
89 1−11.1T+89T2 1 - 11.1T + 89T^{2}
97 1+6.70T+97T2 1 + 6.70T + 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

−8.056733426744796758800390779368, −7.16834339684422802381238351910, −6.80036613043399725917409759551, −6.23292262448878159128716463037, −5.07698801855859205716715204747, −4.04358760341228978143757835957, −3.55835338934226451363465026818, −2.44988023498937485379709386146, −0.937170139549211323205395888021, 0, 0.937170139549211323205395888021, 2.44988023498937485379709386146, 3.55835338934226451363465026818, 4.04358760341228978143757835957, 5.07698801855859205716715204747, 6.23292262448878159128716463037, 6.80036613043399725917409759551, 7.16834339684422802381238351910, 8.056733426744796758800390779368

Graph of the ZZ-function along the critical line