Properties

Label 2-65e2-1.1-c1-0-65
Degree 22
Conductor 42254225
Sign 11
Analytic cond. 33.736733.7367
Root an. cond. 5.808335.80833
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  + 1.21·2-s + 1.31·3-s − 0.525·4-s + 1.59·6-s − 2.90·7-s − 3.06·8-s − 1.28·9-s − 0.214·11-s − 0.688·12-s − 3.52·14-s − 2.67·16-s + 6.42·17-s − 1.55·18-s − 2.21·19-s − 3.80·21-s − 0.260·22-s + 4.68·23-s − 4.02·24-s − 5.61·27-s + 1.52·28-s + 8.70·29-s + 5.59·31-s + 2.88·32-s − 0.280·33-s + 7.80·34-s + 0.673·36-s − 2.28·37-s + ⋯
L(s)  = 1  + 0.858·2-s + 0.756·3-s − 0.262·4-s + 0.649·6-s − 1.09·7-s − 1.08·8-s − 0.426·9-s − 0.0646·11-s − 0.198·12-s − 0.942·14-s − 0.668·16-s + 1.55·17-s − 0.366·18-s − 0.507·19-s − 0.830·21-s − 0.0554·22-s + 0.977·23-s − 0.820·24-s − 1.08·27-s + 0.288·28-s + 1.61·29-s + 1.00·31-s + 0.510·32-s − 0.0489·33-s + 1.33·34-s + 0.112·36-s − 0.374·37-s + ⋯

Functional equation

Λ(s)=(4225s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4225s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 42254225    =    521325^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 33.736733.7367
Root analytic conductor: 5.808335.80833
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4225, ( :1/2), 1)(2,\ 4225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 2.5380410882.538041088
L(12)L(\frac12) \approx 2.5380410882.538041088
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 1 1
13 1 1
good2 11.21T+2T2 1 - 1.21T + 2T^{2}
3 11.31T+3T2 1 - 1.31T + 3T^{2}
7 1+2.90T+7T2 1 + 2.90T + 7T^{2}
11 1+0.214T+11T2 1 + 0.214T + 11T^{2}
17 16.42T+17T2 1 - 6.42T + 17T^{2}
19 1+2.21T+19T2 1 + 2.21T + 19T^{2}
23 14.68T+23T2 1 - 4.68T + 23T^{2}
29 18.70T+29T2 1 - 8.70T + 29T^{2}
31 15.59T+31T2 1 - 5.59T + 31T^{2}
37 1+2.28T+37T2 1 + 2.28T + 37T^{2}
41 1+3.05T+41T2 1 + 3.05T + 41T^{2}
43 16.36T+43T2 1 - 6.36T + 43T^{2}
47 1+1.09T+47T2 1 + 1.09T + 47T^{2}
53 1+6.23T+53T2 1 + 6.23T + 53T^{2}
59 19.26T+59T2 1 - 9.26T + 59T^{2}
61 1+0.280T+61T2 1 + 0.280T + 61T^{2}
67 17.76T+67T2 1 - 7.76T + 67T^{2}
71 16.08T+71T2 1 - 6.08T + 71T^{2}
73 110.2T+73T2 1 - 10.2T + 73T^{2}
79 1+14.2T+79T2 1 + 14.2T + 79T^{2}
83 1+9.52T+83T2 1 + 9.52T + 83T^{2}
89 15.61T+89T2 1 - 5.61T + 89T^{2}
97 118.0T+97T2 1 - 18.0T + 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

−8.515788666886452674183668495883, −7.76700154372687429834130477561, −6.71487164063781241868307298241, −6.12996314920509896819912545903, −5.36784393426314463318916251015, −4.60441166909348913119623895609, −3.56281971992993236525387143163, −3.17615478558072498665926290285, −2.50103964926629481594348090728, −0.74836573588688376833158998024, 0.74836573588688376833158998024, 2.50103964926629481594348090728, 3.17615478558072498665926290285, 3.56281971992993236525387143163, 4.60441166909348913119623895609, 5.36784393426314463318916251015, 6.12996314920509896819912545903, 6.71487164063781241868307298241, 7.76700154372687429834130477561, 8.515788666886452674183668495883

Graph of the ZZ-function along the critical line