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Properties

Label 2-2475-1.1-c3-0-135
Degree 22
Conductor 24752475
Sign −1-1
Analytic cond. 146.029146.029
Root an. cond. 12.084212.0842
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  − 4.03·2-s + 8.31·4-s − 6.49·7-s − 1.27·8-s − 11·11-s + 30.5·13-s + 26.2·14-s − 61.3·16-s + 43.1·17-s − 6.46·19-s + 44.4·22-s − 108.·23-s − 123.·26-s − 54.0·28-s + 274.·29-s − 68.5·31-s + 258.·32-s − 174.·34-s − 402.·37-s + 26.0·38-s + 268.·41-s − 30.5·43-s − 91.4·44-s + 436.·46-s − 31.5·47-s − 300.·49-s + 253.·52-s + ⋯
L(s)  = 1  − 1.42·2-s + 1.03·4-s − 0.350·7-s − 0.0564·8-s − 0.301·11-s + 0.651·13-s + 0.501·14-s − 0.958·16-s + 0.615·17-s − 0.0780·19-s + 0.430·22-s − 0.980·23-s − 0.930·26-s − 0.364·28-s + 1.75·29-s − 0.397·31-s + 1.42·32-s − 0.878·34-s − 1.78·37-s + 0.111·38-s + 1.02·41-s − 0.108·43-s − 0.313·44-s + 1.39·46-s − 0.0980·47-s − 0.876·49-s + 0.677·52-s + ⋯

Functional equation

Λ(s)=(2475s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2475s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24752475    =    32⋅52⋅113^{2} \cdot 5^{2} \cdot 11
Sign: −1-1
Analytic conductor: 146.029146.029
Root analytic conductor: 12.084212.0842
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2475, ( :3/2), −1)(2,\ 2475,\ (\ :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)
bad3 1 1
5 1 1
11 1+11T 1 + 11T
good2 1+4.03T+8T2 1 + 4.03T + 8T^{2}
7 1+6.49T+343T2 1 + 6.49T + 343T^{2}
13 1−30.5T+2.19e3T2 1 - 30.5T + 2.19e3T^{2}
17 1−43.1T+4.91e3T2 1 - 43.1T + 4.91e3T^{2}
19 1+6.46T+6.85e3T2 1 + 6.46T + 6.85e3T^{2}
23 1+108.T+1.21e4T2 1 + 108.T + 1.21e4T^{2}
29 1−274.T+2.43e4T2 1 - 274.T + 2.43e4T^{2}
31 1+68.5T+2.97e4T2 1 + 68.5T + 2.97e4T^{2}
37 1+402.T+5.06e4T2 1 + 402.T + 5.06e4T^{2}
41 1−268.T+6.89e4T2 1 - 268.T + 6.89e4T^{2}
43 1+30.5T+7.95e4T2 1 + 30.5T + 7.95e4T^{2}
47 1+31.5T+1.03e5T2 1 + 31.5T + 1.03e5T^{2}
53 1+252.T+1.48e5T2 1 + 252.T + 1.48e5T^{2}
59 1−558.T+2.05e5T2 1 - 558.T + 2.05e5T^{2}
61 1−335.T+2.26e5T2 1 - 335.T + 2.26e5T^{2}
67 1−28.7T+3.00e5T2 1 - 28.7T + 3.00e5T^{2}
71 1+89.0T+3.57e5T2 1 + 89.0T + 3.57e5T^{2}
73 1+717.T+3.89e5T2 1 + 717.T + 3.89e5T^{2}
79 1−200.T+4.93e5T2 1 - 200.T + 4.93e5T^{2}
83 1+243.T+5.71e5T2 1 + 243.T + 5.71e5T^{2}
89 1−312.T+7.04e5T2 1 - 312.T + 7.04e5T^{2}
97 1−27.9T+9.12e5T2 1 - 27.9T + 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

−8.353269635687547896373284706757, −7.68689843590326600292642929956, −6.85983408587648447115154863048, −6.16322256798716487300231275938, −5.14508688907335366592253377243, −4.08011478932058509051533724222, −3.04222779260184957883982800014, −1.94673506044199708653465687081, −0.997909905006072522647684163779, 0, 0.997909905006072522647684163779, 1.94673506044199708653465687081, 3.04222779260184957883982800014, 4.08011478932058509051533724222, 5.14508688907335366592253377243, 6.16322256798716487300231275938, 6.85983408587648447115154863048, 7.68689843590326600292642929956, 8.353269635687547896373284706757

Graph of the ZZ-function along the critical line