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Properties

Label 2-1575-1.1-c3-0-123
Degree 22
Conductor 15751575
Sign 11
Analytic cond. 92.928092.9280
Root an. cond. 9.639919.63991
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.44·2-s + 21.6·4-s + 7·7-s + 74.3·8-s + 60.4·11-s + 64.0·13-s + 38.1·14-s + 231.·16-s − 37.7·17-s − 134.·19-s + 329.·22-s − 52.6·23-s + 348.·26-s + 151.·28-s + 165.·29-s − 71.9·31-s + 667.·32-s − 205.·34-s + 48.7·37-s − 734.·38-s − 10.5·41-s − 425.·43-s + 1.31e3·44-s − 286.·46-s − 249.·47-s + 49·49-s + 1.38e3·52-s + ⋯
L(s)  = 1  + 1.92·2-s + 2.70·4-s + 0.377·7-s + 3.28·8-s + 1.65·11-s + 1.36·13-s + 0.727·14-s + 3.62·16-s − 0.538·17-s − 1.62·19-s + 3.19·22-s − 0.477·23-s + 2.63·26-s + 1.02·28-s + 1.05·29-s − 0.416·31-s + 3.68·32-s − 1.03·34-s + 0.216·37-s − 3.13·38-s − 0.0402·41-s − 1.51·43-s + 4.48·44-s − 0.918·46-s − 0.773·47-s + 0.142·49-s + 3.70·52-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 92.928092.9280
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1575, ( :3/2), 1)(2,\ 1575,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 10.1504628510.15046285
L(12)L(\frac12) ≈\approx 10.1504628510.15046285
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
7 1−7T 1 - 7T
good2 1−5.44T+8T2 1 - 5.44T + 8T^{2}
11 1−60.4T+1.33e3T2 1 - 60.4T + 1.33e3T^{2}
13 1−64.0T+2.19e3T2 1 - 64.0T + 2.19e3T^{2}
17 1+37.7T+4.91e3T2 1 + 37.7T + 4.91e3T^{2}
19 1+134.T+6.85e3T2 1 + 134.T + 6.85e3T^{2}
23 1+52.6T+1.21e4T2 1 + 52.6T + 1.21e4T^{2}
29 1−165.T+2.43e4T2 1 - 165.T + 2.43e4T^{2}
31 1+71.9T+2.97e4T2 1 + 71.9T + 2.97e4T^{2}
37 1−48.7T+5.06e4T2 1 - 48.7T + 5.06e4T^{2}
41 1+10.5T+6.89e4T2 1 + 10.5T + 6.89e4T^{2}
43 1+425.T+7.95e4T2 1 + 425.T + 7.95e4T^{2}
47 1+249.T+1.03e5T2 1 + 249.T + 1.03e5T^{2}
53 1−544.T+1.48e5T2 1 - 544.T + 1.48e5T^{2}
59 1+567.T+2.05e5T2 1 + 567.T + 2.05e5T^{2}
61 1−614.T+2.26e5T2 1 - 614.T + 2.26e5T^{2}
67 1−201.T+3.00e5T2 1 - 201.T + 3.00e5T^{2}
71 1+525.T+3.57e5T2 1 + 525.T + 3.57e5T^{2}
73 1−137.T+3.89e5T2 1 - 137.T + 3.89e5T^{2}
79 1+234.T+4.93e5T2 1 + 234.T + 4.93e5T^{2}
83 1−10.3T+5.71e5T2 1 - 10.3T + 5.71e5T^{2}
89 1+20.9T+7.04e5T2 1 + 20.9T + 7.04e5T^{2}
97 1−1.38e3T+9.12e5T2 1 - 1.38e3T + 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

−8.856629402957580063645162903067, −8.167243995603459509441097506413, −6.82376220858614218452729041848, −6.48428252222754010003045532916, −5.81304940604708909520005336926, −4.66066281280979298168403826010, −4.07788820175730158385843391977, −3.44230435222000986648953939844, −2.14945308910623565157214415557, −1.34142716982583597837147846290, 1.34142716982583597837147846290, 2.14945308910623565157214415557, 3.44230435222000986648953939844, 4.07788820175730158385843391977, 4.66066281280979298168403826010, 5.81304940604708909520005336926, 6.48428252222754010003045532916, 6.82376220858614218452729041848, 8.167243995603459509441097506413, 8.856629402957580063645162903067

Graph of the ZZ-function along the critical line