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Properties

Label 2-4002-1.1-c1-0-18
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 − 1.56·5-s − 6-s + 1.95·7-s − 8-s + 9-s + 1.56·10-s − 1.08·11-s + 12-s − 0.218·13-s − 1.95·14-s − 1.56·15-s + 16-s + 7.65·17-s − 18-s − 7.93·19-s − 1.56·20-s + 1.95·21-s + 1.08·22-s + 23-s − 24-s − 2.56·25-s + 0.218·26-s + 27-s + 1.95·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.698·5-s − 0.408·6-s + 0.738·7-s − 0.353·8-s + 0.333·9-s + 0.493·10-s − 0.325·11-s + 0.288·12-s − 0.0605·13-s − 0.521·14-s − 0.403·15-s + 0.250·16-s + 1.85·17-s − 0.235·18-s − 1.82·19-s − 0.349·20-s + 0.426·21-s + 0.230·22-s + 0.208·23-s − 0.204·24-s − 0.512·25-s + 0.0428·26-s + 0.192·27-s + 0.369·28-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 1.5627896051.562789605
L(12)L(\frac12) ≈\approx 1.5627896051.562789605
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+1.56T+5T2 1 + 1.56T + 5T^{2}
7 1−1.95T+7T2 1 - 1.95T + 7T^{2}
11 1+1.08T+11T2 1 + 1.08T + 11T^{2}
13 1+0.218T+13T2 1 + 0.218T + 13T^{2}
17 1−7.65T+17T2 1 - 7.65T + 17T^{2}
19 1+7.93T+19T2 1 + 7.93T + 19T^{2}
31 1−3.20T+31T2 1 - 3.20T + 31T^{2}
37 1+0.314T+37T2 1 + 0.314T + 37T^{2}
41 1−11.4T+41T2 1 - 11.4T + 41T^{2}
43 1−1.34T+43T2 1 - 1.34T + 43T^{2}
47 1−6.23T+47T2 1 - 6.23T + 47T^{2}
53 1+8.75T+53T2 1 + 8.75T + 53T^{2}
59 1−12.7T+59T2 1 - 12.7T + 59T^{2}
61 1−5.49T+61T2 1 - 5.49T + 61T^{2}
67 1−3.05T+67T2 1 - 3.05T + 67T^{2}
71 1−5.51T+71T2 1 - 5.51T + 71T^{2}
73 1+9.40T+73T2 1 + 9.40T + 73T^{2}
79 1+12.4T+79T2 1 + 12.4T + 79T^{2}
83 1−4.40T+83T2 1 - 4.40T + 83T^{2}
89 1+1.55T+89T2 1 + 1.55T + 89T^{2}
97 1−10.1T+97T2 1 - 10.1T + 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

−8.356157007317774227264774970488, −7.80743435342889869479355278626, −7.43565541729463868194209663802, −6.39007994534109148126914242761, −5.54247687550545636234077160333, −4.51447147742858893683749105857, −3.79363567087771109781251529602, −2.80978638299004530061618342195, −1.92491183332700487710984118619, −0.78386415290010982350728773879, 0.78386415290010982350728773879, 1.92491183332700487710984118619, 2.80978638299004530061618342195, 3.79363567087771109781251529602, 4.51447147742858893683749105857, 5.54247687550545636234077160333, 6.39007994534109148126914242761, 7.43565541729463868194209663802, 7.80743435342889869479355278626, 8.356157007317774227264774970488

Graph of the ZZ-function along the critical line