↑

Properties

Label 2-1850-1.1-c1-0-50
Degree 22
Conductor 18501850
Sign −1-1
Analytic cond. 14.772314.7723
Root an. cond. 3.843473.84347
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.53·3-s + 4-s − 1.53·6-s + 2.87·7-s + 8-s − 0.630·9-s − 1.09·11-s − 1.53·12-s − 4.53·13-s + 2.87·14-s + 16-s − 2.80·17-s − 0.630·18-s − 5.04·19-s − 4.43·21-s − 1.09·22-s − 7.41·23-s − 1.53·24-s − 4.53·26-s + 5.58·27-s + 2.87·28-s + 6.68·29-s + 3.51·31-s + 32-s + 1.68·33-s − 2.80·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.888·3-s + 0.5·4-s − 0.628·6-s + 1.08·7-s + 0.353·8-s − 0.210·9-s − 0.329·11-s − 0.444·12-s − 1.25·13-s + 0.769·14-s + 0.250·16-s − 0.679·17-s − 0.148·18-s − 1.15·19-s − 0.967·21-s − 0.232·22-s − 1.54·23-s − 0.314·24-s − 0.890·26-s + 1.07·27-s + 0.544·28-s + 1.24·29-s + 0.630·31-s + 0.176·32-s + 0.292·33-s − 0.480·34-s + ⋯

Functional equation

Λ(s)=(1850s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1850s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18501850    =    2⋅52⋅372 \cdot 5^{2} \cdot 37
Sign: −1-1
Analytic conductor: 14.772314.7723
Root analytic conductor: 3.843473.84347
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1850, ( :1/2), −1)(2,\ 1850,\ (\ :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
5 1 1
37 1+T 1 + T
good3 1+1.53T+3T2 1 + 1.53T + 3T^{2}
7 1−2.87T+7T2 1 - 2.87T + 7T^{2}
11 1+1.09T+11T2 1 + 1.09T + 11T^{2}
13 1+4.53T+13T2 1 + 4.53T + 13T^{2}
17 1+2.80T+17T2 1 + 2.80T + 17T^{2}
19 1+5.04T+19T2 1 + 5.04T + 19T^{2}
23 1+7.41T+23T2 1 + 7.41T + 23T^{2}
29 1−6.68T+29T2 1 - 6.68T + 29T^{2}
31 1−3.51T+31T2 1 - 3.51T + 31T^{2}
41 1+8.07T+41T2 1 + 8.07T + 41T^{2}
43 1+10.2T+43T2 1 + 10.2T + 43T^{2}
47 1+8.68T+47T2 1 + 8.68T + 47T^{2}
53 1−10.0T+53T2 1 - 10.0T + 53T^{2}
59 1−10.2T+59T2 1 - 10.2T + 59T^{2}
61 1−6.29T+61T2 1 - 6.29T + 61T^{2}
67 1+13.2T+67T2 1 + 13.2T + 67T^{2}
71 1−6.29T+71T2 1 - 6.29T + 71T^{2}
73 1+12.7T+73T2 1 + 12.7T + 73T^{2}
79 1−2.58T+79T2 1 - 2.58T + 79T^{2}
83 1+8.48T+83T2 1 + 8.48T + 83T^{2}
89 1+6.51T+89T2 1 + 6.51T + 89T^{2}
97 1+3.07T+97T2 1 + 3.07T + 97T^{2}
show more
show less
   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

−8.494031857496862585393605755609, −8.166922527811166607370359984698, −6.96556838448942531042278721785, −6.38647032776418567834371524018, −5.41035212151330347280175613963, −4.84772763250043229420889753281, −4.22090296842033137633880042530, −2.73270499268890823377421484147, −1.83267338175141695976612589082, 0, 1.83267338175141695976612589082, 2.73270499268890823377421484147, 4.22090296842033137633880042530, 4.84772763250043229420889753281, 5.41035212151330347280175613963, 6.38647032776418567834371524018, 6.96556838448942531042278721785, 8.166922527811166607370359984698, 8.494031857496862585393605755609

Graph of the ZZ-function along the critical line