Properties

Label 2-4805-1.1-c1-0-15
Degree 22
Conductor 48054805
Sign 11
Analytic cond. 38.368138.3681
Root an. cond. 6.194206.19420
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  − 1.35·2-s + 0.880·3-s − 0.162·4-s + 5-s − 1.19·6-s − 3.54·7-s + 2.93·8-s − 2.22·9-s − 1.35·10-s − 6.04·11-s − 0.142·12-s − 0.780·13-s + 4.81·14-s + 0.880·15-s − 3.64·16-s − 3.48·17-s + 3.01·18-s − 4.59·19-s − 0.162·20-s − 3.12·21-s + 8.19·22-s + 2.96·23-s + 2.58·24-s + 25-s + 1.05·26-s − 4.59·27-s + 0.575·28-s + ⋯
L(s)  = 1  − 0.958·2-s + 0.508·3-s − 0.0810·4-s + 0.447·5-s − 0.487·6-s − 1.34·7-s + 1.03·8-s − 0.741·9-s − 0.428·10-s − 1.82·11-s − 0.0412·12-s − 0.216·13-s + 1.28·14-s + 0.227·15-s − 0.912·16-s − 0.844·17-s + 0.710·18-s − 1.05·19-s − 0.0362·20-s − 0.681·21-s + 1.74·22-s + 0.619·23-s + 0.526·24-s + 0.200·25-s + 0.207·26-s − 0.885·27-s + 0.108·28-s + ⋯

Functional equation

Λ(s)=(4805s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4805s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 48054805    =    53125 \cdot 31^{2}
Sign: 11
Analytic conductor: 38.368138.3681
Root analytic conductor: 6.194206.19420
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4805, ( :1/2), 1)(2,\ 4805,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 0.25075664250.2507566425
L(12)L(\frac12) \approx 0.25075664250.2507566425
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1T 1 - T
31 1 1
good2 1+1.35T+2T2 1 + 1.35T + 2T^{2}
3 10.880T+3T2 1 - 0.880T + 3T^{2}
7 1+3.54T+7T2 1 + 3.54T + 7T^{2}
11 1+6.04T+11T2 1 + 6.04T + 11T^{2}
13 1+0.780T+13T2 1 + 0.780T + 13T^{2}
17 1+3.48T+17T2 1 + 3.48T + 17T^{2}
19 1+4.59T+19T2 1 + 4.59T + 19T^{2}
23 12.96T+23T2 1 - 2.96T + 23T^{2}
29 15.27T+29T2 1 - 5.27T + 29T^{2}
37 1+4.63T+37T2 1 + 4.63T + 37T^{2}
41 1+7.86T+41T2 1 + 7.86T + 41T^{2}
43 10.825T+43T2 1 - 0.825T + 43T^{2}
47 1+11.6T+47T2 1 + 11.6T + 47T^{2}
53 1+9.04T+53T2 1 + 9.04T + 53T^{2}
59 112.3T+59T2 1 - 12.3T + 59T^{2}
61 1+1.81T+61T2 1 + 1.81T + 61T^{2}
67 1+7.06T+67T2 1 + 7.06T + 67T^{2}
71 115.7T+71T2 1 - 15.7T + 71T^{2}
73 11.06T+73T2 1 - 1.06T + 73T^{2}
79 1+12.1T+79T2 1 + 12.1T + 79T^{2}
83 111.8T+83T2 1 - 11.8T + 83T^{2}
89 1+3.40T+89T2 1 + 3.40T + 89T^{2}
97 1+5.51T+97T2 1 + 5.51T + 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−8.472304535079587586798103479336, −7.85975606883891755581623996192, −6.90781242163075747140380259654, −6.35313913080618449420902668362, −5.30392541679114789270573352950, −4.70809712963102967872389005424, −3.44243117071066128575369892462, −2.72070290754976200306725100819, −2.00269884508904230506046073489, −0.28867199182917198821895248717, 0.28867199182917198821895248717, 2.00269884508904230506046073489, 2.72070290754976200306725100819, 3.44243117071066128575369892462, 4.70809712963102967872389005424, 5.30392541679114789270573352950, 6.35313913080618449420902668362, 6.90781242163075747140380259654, 7.85975606883891755581623996192, 8.472304535079587586798103479336

Graph of the ZZ-function along the critical line