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Properties

Label 2-7600-1.1-c1-0-161
Degree 22
Conductor 76007600
Sign −1-1
Analytic cond. 60.686360.6863
Root an. cond. 7.790147.79014
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.03·3-s − 2.46·7-s + 6.19·9-s − 0.728·11-s − 6.23·13-s + 0.563·17-s + 19-s − 7.49·21-s − 4.63·23-s + 9.70·27-s + 10.2·29-s − 6.06·31-s − 2.21·33-s − 5.72·37-s − 18.9·39-s + 4.79·41-s − 8.06·43-s − 8.12·47-s − 0.900·49-s + 1.70·51-s − 1.53·53-s + 3.03·57-s + 5.76·59-s + 10.9·61-s − 15.3·63-s − 12.9·67-s − 14.0·69-s + ⋯
L(s)  = 1  + 1.75·3-s − 0.933·7-s + 2.06·9-s − 0.219·11-s − 1.72·13-s + 0.136·17-s + 0.229·19-s − 1.63·21-s − 0.966·23-s + 1.86·27-s + 1.89·29-s − 1.08·31-s − 0.384·33-s − 0.941·37-s − 3.02·39-s + 0.748·41-s − 1.23·43-s − 1.18·47-s − 0.128·49-s + 0.239·51-s − 0.210·53-s + 0.401·57-s + 0.750·59-s + 1.40·61-s − 1.92·63-s − 1.58·67-s − 1.69·69-s + ⋯

Functional equation

Λ(s)=(7600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76007600    =    24⋅52⋅192^{4} \cdot 5^{2} \cdot 19
Sign: −1-1
Analytic conductor: 60.686360.6863
Root analytic conductor: 7.790147.79014
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7600, ( :1/2), −1)(2,\ 7600,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5 1 1
19 1−T 1 - T
good3 1−3.03T+3T2 1 - 3.03T + 3T^{2}
7 1+2.46T+7T2 1 + 2.46T + 7T^{2}
11 1+0.728T+11T2 1 + 0.728T + 11T^{2}
13 1+6.23T+13T2 1 + 6.23T + 13T^{2}
17 1−0.563T+17T2 1 - 0.563T + 17T^{2}
23 1+4.63T+23T2 1 + 4.63T + 23T^{2}
29 1−10.2T+29T2 1 - 10.2T + 29T^{2}
31 1+6.06T+31T2 1 + 6.06T + 31T^{2}
37 1+5.72T+37T2 1 + 5.72T + 37T^{2}
41 1−4.79T+41T2 1 - 4.79T + 41T^{2}
43 1+8.06T+43T2 1 + 8.06T + 43T^{2}
47 1+8.12T+47T2 1 + 8.12T + 47T^{2}
53 1+1.53T+53T2 1 + 1.53T + 53T^{2}
59 1−5.76T+59T2 1 - 5.76T + 59T^{2}
61 1−10.9T+61T2 1 - 10.9T + 61T^{2}
67 1+12.9T+67T2 1 + 12.9T + 67T^{2}
71 1−4.39T+71T2 1 - 4.39T + 71T^{2}
73 1+4.09T+73T2 1 + 4.09T + 73T^{2}
79 1+15.3T+79T2 1 + 15.3T + 79T^{2}
83 1+7.85T+83T2 1 + 7.85T + 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1−11.0T+97T2 1 - 11.0T + 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

−7.54473996115085202949537690601, −7.09765538596934571176345506854, −6.39822803016792543992216528635, −5.27411513654731338667265850216, −4.51576829712980586011434820404, −3.69347592307226967825423648215, −2.98386071276938996502637503944, −2.51662657872880908752851396783, −1.62843825262510491992718531156, 0, 1.62843825262510491992718531156, 2.51662657872880908752851396783, 2.98386071276938996502637503944, 3.69347592307226967825423648215, 4.51576829712980586011434820404, 5.27411513654731338667265850216, 6.39822803016792543992216528635, 7.09765538596934571176345506854, 7.54473996115085202949537690601

Graph of the ZZ-function along the critical line