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Properties

Label 2-5175-1.1-c1-0-168
Degree 22
Conductor 51755175
Sign −1-1
Analytic cond. 41.322541.3225
Root an. cond. 6.428266.42826
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  + 2.23·2-s + 3.00·4-s + 1.23·7-s + 2.23·8-s − 4·11-s − 4.47·13-s + 2.76·14-s − 0.999·16-s − 7.23·17-s + 2.76·19-s − 8.94·22-s + 23-s − 10.0·26-s + 3.70·28-s + 4.47·29-s + 2.47·31-s − 6.70·32-s − 16.1·34-s + 4.47·37-s + 6.18·38-s − 6.94·41-s − 7.70·43-s − 12.0·44-s + 2.23·46-s − 4·47-s − 5.47·49-s − 13.4·52-s + ⋯
L(s)  = 1  + 1.58·2-s + 1.50·4-s + 0.467·7-s + 0.790·8-s − 1.20·11-s − 1.24·13-s + 0.738·14-s − 0.249·16-s − 1.75·17-s + 0.634·19-s − 1.90·22-s + 0.208·23-s − 1.96·26-s + 0.700·28-s + 0.830·29-s + 0.444·31-s − 1.18·32-s − 2.77·34-s + 0.735·37-s + 1.00·38-s − 1.08·41-s − 1.17·43-s − 1.80·44-s + 0.329·46-s − 0.583·47-s − 0.781·49-s − 1.86·52-s + ⋯

Functional equation

Λ(s)=(5175s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5175s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 51755175    =    32⋅52⋅233^{2} \cdot 5^{2} \cdot 23
Sign: −1-1
Analytic conductor: 41.322541.3225
Root analytic conductor: 6.428266.42826
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5175, ( :1/2), −1)(2,\ 5175,\ (\ :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)
bad3 1 1
5 1 1
23 1−T 1 - T
good2 1−2.23T+2T2 1 - 2.23T + 2T^{2}
7 1−1.23T+7T2 1 - 1.23T + 7T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1+4.47T+13T2 1 + 4.47T + 13T^{2}
17 1+7.23T+17T2 1 + 7.23T + 17T^{2}
19 1−2.76T+19T2 1 - 2.76T + 19T^{2}
29 1−4.47T+29T2 1 - 4.47T + 29T^{2}
31 1−2.47T+31T2 1 - 2.47T + 31T^{2}
37 1−4.47T+37T2 1 - 4.47T + 37T^{2}
41 1+6.94T+41T2 1 + 6.94T + 41T^{2}
43 1+7.70T+43T2 1 + 7.70T + 43T^{2}
47 1+4T+47T2 1 + 4T + 47T^{2}
53 1+0.763T+53T2 1 + 0.763T + 53T^{2}
59 1+12.9T+59T2 1 + 12.9T + 59T^{2}
61 1+4.47T+61T2 1 + 4.47T + 61T^{2}
67 1+5.23T+67T2 1 + 5.23T + 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1−10.9T+73T2 1 - 10.9T + 73T^{2}
79 1+3.70T+79T2 1 + 3.70T + 79T^{2}
83 1−4T+83T2 1 - 4T + 83T^{2}
89 1+3.23T+89T2 1 + 3.23T + 89T^{2}
97 1−0.472T+97T2 1 - 0.472T + 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

−7.68478877897372664592891740590, −6.86187893238974572079922039142, −6.34348417438734229098535940613, −5.33712232661972509256012598326, −4.77619728132496108747145644917, −4.53891000696561703427098557246, −3.26811208157143016797753365659, −2.65209461058203472766778692329, −1.90726276821360446601291293866, 0, 1.90726276821360446601291293866, 2.65209461058203472766778692329, 3.26811208157143016797753365659, 4.53891000696561703427098557246, 4.77619728132496108747145644917, 5.33712232661972509256012598326, 6.34348417438734229098535940613, 6.86187893238974572079922039142, 7.68478877897372664592891740590

Graph of the ZZ-function along the critical line