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Properties

Label 2-1152-1.1-c3-0-35
Degree 22
Conductor 11521152
Sign −1-1
Analytic cond. 67.970267.9702
Root an. cond. 8.244408.24440
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  − 18.7·5-s + 19.7·7-s − 5.84·11-s − 33.7·13-s − 33.2·17-s + 102.·19-s + 130.·23-s + 228.·25-s − 111.·29-s + 257.·31-s − 370.·35-s − 424.·37-s + 10.9·41-s + 324.·43-s − 599.·47-s + 45.7·49-s − 106.·53-s + 109.·55-s + 617.·59-s + 325.·61-s + 634.·65-s − 240.·67-s + 123.·71-s − 46.7·73-s − 115.·77-s − 547.·79-s − 1.21e3·83-s + ⋯
L(s)  = 1  − 1.68·5-s + 1.06·7-s − 0.160·11-s − 0.719·13-s − 0.474·17-s + 1.24·19-s + 1.17·23-s + 1.82·25-s − 0.715·29-s + 1.49·31-s − 1.78·35-s − 1.88·37-s + 0.0416·41-s + 1.15·43-s − 1.86·47-s + 0.133·49-s − 0.276·53-s + 0.269·55-s + 1.36·59-s + 0.682·61-s + 1.21·65-s − 0.437·67-s + 0.207·71-s − 0.0749·73-s − 0.170·77-s − 0.779·79-s − 1.60·83-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: −1-1
Analytic conductor: 67.970267.9702
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1152, ( :3/2), −1)(2,\ 1152,\ (\ :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)
bad2 1 1
3 1 1
good5 1+18.7T+125T2 1 + 18.7T + 125T^{2}
7 1−19.7T+343T2 1 - 19.7T + 343T^{2}
11 1+5.84T+1.33e3T2 1 + 5.84T + 1.33e3T^{2}
13 1+33.7T+2.19e3T2 1 + 33.7T + 2.19e3T^{2}
17 1+33.2T+4.91e3T2 1 + 33.2T + 4.91e3T^{2}
19 1−102.T+6.85e3T2 1 - 102.T + 6.85e3T^{2}
23 1−130.T+1.21e4T2 1 - 130.T + 1.21e4T^{2}
29 1+111.T+2.43e4T2 1 + 111.T + 2.43e4T^{2}
31 1−257.T+2.97e4T2 1 - 257.T + 2.97e4T^{2}
37 1+424.T+5.06e4T2 1 + 424.T + 5.06e4T^{2}
41 1−10.9T+6.89e4T2 1 - 10.9T + 6.89e4T^{2}
43 1−324.T+7.95e4T2 1 - 324.T + 7.95e4T^{2}
47 1+599.T+1.03e5T2 1 + 599.T + 1.03e5T^{2}
53 1+106.T+1.48e5T2 1 + 106.T + 1.48e5T^{2}
59 1−617.T+2.05e5T2 1 - 617.T + 2.05e5T^{2}
61 1−325.T+2.26e5T2 1 - 325.T + 2.26e5T^{2}
67 1+240.T+3.00e5T2 1 + 240.T + 3.00e5T^{2}
71 1−123.T+3.57e5T2 1 - 123.T + 3.57e5T^{2}
73 1+46.7T+3.89e5T2 1 + 46.7T + 3.89e5T^{2}
79 1+547.T+4.93e5T2 1 + 547.T + 4.93e5T^{2}
83 1+1.21e3T+5.71e5T2 1 + 1.21e3T + 5.71e5T^{2}
89 1−1.62e3T+7.04e5T2 1 - 1.62e3T + 7.04e5T^{2}
97 1+1.27e3T+9.12e5T2 1 + 1.27e3T + 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

−8.752893435434179807092589020598, −8.128492161959012685480570956475, −7.44063718675487939688833999677, −6.86185151681756448688125697595, −5.22649852598562012707837008044, −4.73261829764607522386804704492, −3.75565105801924994135118994312, −2.76578288630528469667021432288, −1.23412946772039298846437281409, 0, 1.23412946772039298846437281409, 2.76578288630528469667021432288, 3.75565105801924994135118994312, 4.73261829764607522386804704492, 5.22649852598562012707837008044, 6.86185151681756448688125697595, 7.44063718675487939688833999677, 8.128492161959012685480570956475, 8.752893435434179807092589020598

Graph of the ZZ-function along the critical line