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Properties

Label 2-1859-1.1-c3-0-385
Degree 22
Conductor 18591859
Sign −1-1
Analytic cond. 109.684109.684
Root an. cond. 10.473010.4730
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.85·2-s + 3.92·3-s + 15.5·4-s + 7.28·5-s + 19.0·6-s − 33.6·7-s + 36.5·8-s − 11.5·9-s + 35.3·10-s − 11·11-s + 61.0·12-s − 163.·14-s + 28.6·15-s + 53.2·16-s + 19.6·17-s − 56.1·18-s − 140.·19-s + 113.·20-s − 132.·21-s − 53.3·22-s + 49.5·23-s + 143.·24-s − 71.9·25-s − 151.·27-s − 522.·28-s − 51.1·29-s + 138.·30-s + ⋯
L(s)  = 1  + 1.71·2-s + 0.755·3-s + 1.94·4-s + 0.651·5-s + 1.29·6-s − 1.81·7-s + 1.61·8-s − 0.428·9-s + 1.11·10-s − 0.301·11-s + 1.46·12-s − 3.11·14-s + 0.492·15-s + 0.831·16-s + 0.280·17-s − 0.735·18-s − 1.69·19-s + 1.26·20-s − 1.37·21-s − 0.517·22-s + 0.449·23-s + 1.22·24-s − 0.575·25-s − 1.07·27-s − 3.52·28-s − 0.327·29-s + 0.844·30-s + ⋯

Functional equation

Λ(s)=(1859s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1859s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18591859    =    11⋅13211 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 109.684109.684
Root analytic conductor: 10.473010.4730
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1859, ( :3/2), −1)(2,\ 1859,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad11 1+11T 1 + 11T
13 1 1
good2 1−4.85T+8T2 1 - 4.85T + 8T^{2}
3 1−3.92T+27T2 1 - 3.92T + 27T^{2}
5 1−7.28T+125T2 1 - 7.28T + 125T^{2}
7 1+33.6T+343T2 1 + 33.6T + 343T^{2}
17 1−19.6T+4.91e3T2 1 - 19.6T + 4.91e3T^{2}
19 1+140.T+6.85e3T2 1 + 140.T + 6.85e3T^{2}
23 1−49.5T+1.21e4T2 1 - 49.5T + 1.21e4T^{2}
29 1+51.1T+2.43e4T2 1 + 51.1T + 2.43e4T^{2}
31 1+28.5T+2.97e4T2 1 + 28.5T + 2.97e4T^{2}
37 1+157.T+5.06e4T2 1 + 157.T + 5.06e4T^{2}
41 1+253.T+6.89e4T2 1 + 253.T + 6.89e4T^{2}
43 1+37.2T+7.95e4T2 1 + 37.2T + 7.95e4T^{2}
47 1−579.T+1.03e5T2 1 - 579.T + 1.03e5T^{2}
53 1−412.T+1.48e5T2 1 - 412.T + 1.48e5T^{2}
59 1+423.T+2.05e5T2 1 + 423.T + 2.05e5T^{2}
61 1−249.T+2.26e5T2 1 - 249.T + 2.26e5T^{2}
67 1−635.T+3.00e5T2 1 - 635.T + 3.00e5T^{2}
71 1−718.T+3.57e5T2 1 - 718.T + 3.57e5T^{2}
73 1+1.15e3T+3.89e5T2 1 + 1.15e3T + 3.89e5T^{2}
79 1+388.T+4.93e5T2 1 + 388.T + 4.93e5T^{2}
83 1+616.T+5.71e5T2 1 + 616.T + 5.71e5T^{2}
89 1−1.05e3T+7.04e5T2 1 - 1.05e3T + 7.04e5T^{2}
97 1+1.11e3T+9.12e5T2 1 + 1.11e3T + 9.12e5T^{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

−8.585606803856572552039672144696, −7.31190284599908324800697431804, −6.57398895564289668008456436248, −5.94997517312382934606905342331, −5.37650160826345035061269418435, −4.07613073572611803752262068647, −3.47912525211879763783642193432, −2.69277052799655231740459918070, −2.10806371674581315788656654766, 0, 2.10806371674581315788656654766, 2.69277052799655231740459918070, 3.47912525211879763783642193432, 4.07613073572611803752262068647, 5.37650160826345035061269418435, 5.94997517312382934606905342331, 6.57398895564289668008456436248, 7.31190284599908324800697431804, 8.585606803856572552039672144696

Graph of the ZZ-function along the critical line