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Properties

Label 2-4608-1.1-c1-0-6
Degree 22
Conductor 46084608
Sign 11
Analytic cond. 36.795036.7950
Root an. cond. 6.065896.06589
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  − 2.44·5-s − 2.44·7-s + 2·11-s − 3.46·13-s + 2.82·17-s + 2.82·19-s − 6.92·23-s + 0.999·25-s − 2.44·29-s − 7.34·31-s + 5.99·35-s + 10.3·37-s − 8.48·41-s − 2.82·43-s + 6.92·47-s − 1.00·49-s − 2.44·53-s − 4.89·55-s + 8·59-s − 3.46·61-s + 8.48·65-s − 11.3·67-s + 13.8·71-s − 4.89·77-s − 2.44·79-s + 14·83-s − 6.92·85-s + ⋯
L(s)  = 1  − 1.09·5-s − 0.925·7-s + 0.603·11-s − 0.960·13-s + 0.685·17-s + 0.648·19-s − 1.44·23-s + 0.199·25-s − 0.454·29-s − 1.31·31-s + 1.01·35-s + 1.70·37-s − 1.32·41-s − 0.431·43-s + 1.01·47-s − 0.142·49-s − 0.336·53-s − 0.660·55-s + 1.04·59-s − 0.443·61-s + 1.05·65-s − 1.38·67-s + 1.64·71-s − 0.558·77-s − 0.275·79-s + 1.53·83-s − 0.751·85-s + ⋯

Functional equation

Λ(s)=(4608s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4608s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46084608    =    29⋅322^{9} \cdot 3^{2}
Sign: 11
Analytic conductor: 36.795036.7950
Root analytic conductor: 6.065896.06589
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4608, ( :1/2), 1)(2,\ 4608,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.85245380950.8524538095
L(12)L(\frac12) ≈\approx 0.85245380950.8524538095
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 1
3 1 1
good5 1+2.44T+5T2 1 + 2.44T + 5T^{2}
7 1+2.44T+7T2 1 + 2.44T + 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
13 1+3.46T+13T2 1 + 3.46T + 13T^{2}
17 1−2.82T+17T2 1 - 2.82T + 17T^{2}
19 1−2.82T+19T2 1 - 2.82T + 19T^{2}
23 1+6.92T+23T2 1 + 6.92T + 23T^{2}
29 1+2.44T+29T2 1 + 2.44T + 29T^{2}
31 1+7.34T+31T2 1 + 7.34T + 31T^{2}
37 1−10.3T+37T2 1 - 10.3T + 37T^{2}
41 1+8.48T+41T2 1 + 8.48T + 41T^{2}
43 1+2.82T+43T2 1 + 2.82T + 43T^{2}
47 1−6.92T+47T2 1 - 6.92T + 47T^{2}
53 1+2.44T+53T2 1 + 2.44T + 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1+3.46T+61T2 1 + 3.46T + 61T^{2}
67 1+11.3T+67T2 1 + 11.3T + 67T^{2}
71 1−13.8T+71T2 1 - 13.8T + 71T^{2}
73 1+73T2 1 + 73T^{2}
79 1+2.44T+79T2 1 + 2.44T + 79T^{2}
83 1−14T+83T2 1 - 14T + 83T^{2}
89 1+11.3T+89T2 1 + 11.3T + 89T^{2}
97 1+6T+97T2 1 + 6T + 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

−8.126321365908853104770171497943, −7.59252219726227429537947139227, −7.02867947761959054988494369536, −6.16194339391290372848901710076, −5.43205291142976141880303332137, −4.41170829038614759696285665513, −3.71132782196901321564023521387, −3.14650771664244245039776594631, −1.95830017725136299138067411900, −0.49496679606924744206973904882, 0.49496679606924744206973904882, 1.95830017725136299138067411900, 3.14650771664244245039776594631, 3.71132782196901321564023521387, 4.41170829038614759696285665513, 5.43205291142976141880303332137, 6.16194339391290372848901710076, 7.02867947761959054988494369536, 7.59252219726227429537947139227, 8.126321365908853104770171497943

Graph of the ZZ-function along the critical line