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Properties

Label 2-9405-1.1-c1-0-134
Degree 22
Conductor 94059405
Sign 11
Analytic cond. 75.099375.0993
Root an. cond. 8.665988.66598
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.39·2-s + 3.74·4-s − 5-s − 2.89·7-s + 4.19·8-s − 2.39·10-s + 11-s + 4.73·13-s − 6.93·14-s + 2.56·16-s + 5.65·17-s + 19-s − 3.74·20-s + 2.39·22-s + 4.00·23-s + 25-s + 11.3·26-s − 10.8·28-s − 9.32·29-s − 6.60·31-s − 2.24·32-s + 13.5·34-s + 2.89·35-s + 6.07·37-s + 2.39·38-s − 4.19·40-s − 5.47·41-s + ⋯
L(s)  = 1  + 1.69·2-s + 1.87·4-s − 0.447·5-s − 1.09·7-s + 1.48·8-s − 0.758·10-s + 0.301·11-s + 1.31·13-s − 1.85·14-s + 0.640·16-s + 1.37·17-s + 0.229·19-s − 0.838·20-s + 0.511·22-s + 0.835·23-s + 0.200·25-s + 2.22·26-s − 2.05·28-s − 1.73·29-s − 1.18·31-s − 0.397·32-s + 2.32·34-s + 0.489·35-s + 0.999·37-s + 0.388·38-s − 0.663·40-s − 0.854·41-s + ⋯

Functional equation

Λ(s)=(9405s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9405s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 94059405    =    32⋅5⋅11⋅193^{2} \cdot 5 \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 75.099375.0993
Root analytic conductor: 8.665988.66598
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9405, ( :1/2), 1)(2,\ 9405,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.4477188745.447718874
L(12)L(\frac12) ≈\approx 5.4477188745.447718874
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)
bad3 1 1
5 1+T 1 + T
11 1−T 1 - T
19 1−T 1 - T
good2 1−2.39T+2T2 1 - 2.39T + 2T^{2}
7 1+2.89T+7T2 1 + 2.89T + 7T^{2}
13 1−4.73T+13T2 1 - 4.73T + 13T^{2}
17 1−5.65T+17T2 1 - 5.65T + 17T^{2}
23 1−4.00T+23T2 1 - 4.00T + 23T^{2}
29 1+9.32T+29T2 1 + 9.32T + 29T^{2}
31 1+6.60T+31T2 1 + 6.60T + 31T^{2}
37 1−6.07T+37T2 1 - 6.07T + 37T^{2}
41 1+5.47T+41T2 1 + 5.47T + 41T^{2}
43 1−10.9T+43T2 1 - 10.9T + 43T^{2}
47 1−0.295T+47T2 1 - 0.295T + 47T^{2}
53 1−3.81T+53T2 1 - 3.81T + 53T^{2}
59 1−5.54T+59T2 1 - 5.54T + 59T^{2}
61 1+1.01T+61T2 1 + 1.01T + 61T^{2}
67 1−6.98T+67T2 1 - 6.98T + 67T^{2}
71 1−1.02T+71T2 1 - 1.02T + 71T^{2}
73 1+0.202T+73T2 1 + 0.202T + 73T^{2}
79 1−7.28T+79T2 1 - 7.28T + 79T^{2}
83 1−13.7T+83T2 1 - 13.7T + 83T^{2}
89 1−15.8T+89T2 1 - 15.8T + 89T^{2}
97 1−4.81T+97T2 1 - 4.81T + 97T^{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

−7.33908764996712674220205317201, −6.87204956272729777408502061590, −5.99287652759232128939669207394, −5.74988290362478611505772049188, −4.96128856330064712339818867161, −3.92940805541184765348540977539, −3.57190785743125325960999159437, −3.16317972728223342266884101245, −2.07283915293441647267145009974, −0.869743060948438292399171644980, 0.869743060948438292399171644980, 2.07283915293441647267145009974, 3.16317972728223342266884101245, 3.57190785743125325960999159437, 3.92940805541184765348540977539, 4.96128856330064712339818867161, 5.74988290362478611505772049188, 5.99287652759232128939669207394, 6.87204956272729777408502061590, 7.33908764996712674220205317201

Graph of the ZZ-function along the critical line