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Properties

Label 2-1859-1.1-c3-0-104
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  − 4.68·2-s − 2.64·3-s + 13.9·4-s − 16.8·5-s + 12.3·6-s − 13.2·7-s − 28.0·8-s − 20.0·9-s + 79.2·10-s − 11·11-s − 36.9·12-s + 62.3·14-s + 44.6·15-s + 19.6·16-s − 38.1·17-s + 93.8·18-s − 48.0·19-s − 236.·20-s + 35.1·21-s + 51.5·22-s − 161.·23-s + 74.1·24-s + 160.·25-s + 124.·27-s − 185.·28-s + 303.·29-s − 209.·30-s + ⋯
L(s)  = 1  − 1.65·2-s − 0.508·3-s + 1.74·4-s − 1.51·5-s + 0.842·6-s − 0.717·7-s − 1.23·8-s − 0.741·9-s + 2.50·10-s − 0.301·11-s − 0.888·12-s + 1.18·14-s + 0.768·15-s + 0.307·16-s − 0.544·17-s + 1.22·18-s − 0.579·19-s − 2.64·20-s + 0.364·21-s + 0.499·22-s − 1.46·23-s + 0.630·24-s + 1.28·25-s + 0.885·27-s − 1.25·28-s + 1.94·29-s − 1.27·30-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+4.68T+8T2 1 + 4.68T + 8T^{2}
3 1+2.64T+27T2 1 + 2.64T + 27T^{2}
5 1+16.8T+125T2 1 + 16.8T + 125T^{2}
7 1+13.2T+343T2 1 + 13.2T + 343T^{2}
17 1+38.1T+4.91e3T2 1 + 38.1T + 4.91e3T^{2}
19 1+48.0T+6.85e3T2 1 + 48.0T + 6.85e3T^{2}
23 1+161.T+1.21e4T2 1 + 161.T + 1.21e4T^{2}
29 1−303.T+2.43e4T2 1 - 303.T + 2.43e4T^{2}
31 1+261.T+2.97e4T2 1 + 261.T + 2.97e4T^{2}
37 1+405.T+5.06e4T2 1 + 405.T + 5.06e4T^{2}
41 1+241.T+6.89e4T2 1 + 241.T + 6.89e4T^{2}
43 1−184.T+7.95e4T2 1 - 184.T + 7.95e4T^{2}
47 1+80.8T+1.03e5T2 1 + 80.8T + 1.03e5T^{2}
53 1−472.T+1.48e5T2 1 - 472.T + 1.48e5T^{2}
59 1−260.T+2.05e5T2 1 - 260.T + 2.05e5T^{2}
61 1−336.T+2.26e5T2 1 - 336.T + 2.26e5T^{2}
67 1−417.T+3.00e5T2 1 - 417.T + 3.00e5T^{2}
71 1−140.T+3.57e5T2 1 - 140.T + 3.57e5T^{2}
73 1−496.T+3.89e5T2 1 - 496.T + 3.89e5T^{2}
79 1−120.T+4.93e5T2 1 - 120.T + 4.93e5T^{2}
83 1−561.T+5.71e5T2 1 - 561.T + 5.71e5T^{2}
89 1+607.T+7.04e5T2 1 + 607.T + 7.04e5T^{2}
97 1+49.5T+9.12e5T2 1 + 49.5T + 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

−8.293341729528764606632691769648, −8.147251570099646170852320142856, −6.93971870827145390028593656980, −6.64383510032891576337090054076, −5.43449856542786442036287346918, −4.21683965392705263648634953212, −3.24097917025196583994852518986, −2.12858640506250261780818760002, −0.56363589350013418841133830134, 0, 0.56363589350013418841133830134, 2.12858640506250261780818760002, 3.24097917025196583994852518986, 4.21683965392705263648634953212, 5.43449856542786442036287346918, 6.64383510032891576337090054076, 6.93971870827145390028593656980, 8.147251570099646170852320142856, 8.293341729528764606632691769648

Graph of the ZZ-function along the critical line