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Properties

Label 2-85e2-1.1-c1-0-339
Degree 22
Conductor 72257225
Sign −1-1
Analytic cond. 57.691957.6919
Root an. cond. 7.595517.59551
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  + 2.10·2-s − 1.80·3-s + 2.43·4-s − 3.80·6-s + 2.68·7-s + 0.906·8-s + 0.260·9-s − 0.975·11-s − 4.38·12-s − 2.65·13-s + 5.64·14-s − 2.95·16-s + 0.547·18-s + 0.676·19-s − 4.84·21-s − 2.05·22-s + 4.03·23-s − 1.63·24-s − 5.58·26-s + 4.94·27-s + 6.52·28-s + 10.0·29-s − 9.85·31-s − 8.02·32-s + 1.76·33-s + 0.632·36-s − 6.83·37-s + ⋯
L(s)  = 1  + 1.48·2-s − 1.04·3-s + 1.21·4-s − 1.55·6-s + 1.01·7-s + 0.320·8-s + 0.0866·9-s − 0.294·11-s − 1.26·12-s − 0.735·13-s + 1.50·14-s − 0.738·16-s + 0.129·18-s + 0.155·19-s − 1.05·21-s − 0.437·22-s + 0.841·23-s − 0.334·24-s − 1.09·26-s + 0.952·27-s + 1.23·28-s + 1.86·29-s − 1.77·31-s − 1.41·32-s + 0.306·33-s + 0.105·36-s − 1.12·37-s + ⋯

Functional equation

Λ(s)=(7225s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7225s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 72257225    =    52⋅1725^{2} \cdot 17^{2}
Sign: −1-1
Analytic conductor: 57.691957.6919
Root analytic conductor: 7.595517.59551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7225, ( :1/2), −1)(2,\ 7225,\ (\ :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)
bad5 1 1
17 1 1
good2 1−2.10T+2T2 1 - 2.10T + 2T^{2}
3 1+1.80T+3T2 1 + 1.80T + 3T^{2}
7 1−2.68T+7T2 1 - 2.68T + 7T^{2}
11 1+0.975T+11T2 1 + 0.975T + 11T^{2}
13 1+2.65T+13T2 1 + 2.65T + 13T^{2}
19 1−0.676T+19T2 1 - 0.676T + 19T^{2}
23 1−4.03T+23T2 1 - 4.03T + 23T^{2}
29 1−10.0T+29T2 1 - 10.0T + 29T^{2}
31 1+9.85T+31T2 1 + 9.85T + 31T^{2}
37 1+6.83T+37T2 1 + 6.83T + 37T^{2}
41 1−7.32T+41T2 1 - 7.32T + 41T^{2}
43 1+8.90T+43T2 1 + 8.90T + 43T^{2}
47 1+0.874T+47T2 1 + 0.874T + 47T^{2}
53 1−2.84T+53T2 1 - 2.84T + 53T^{2}
59 1+10.7T+59T2 1 + 10.7T + 59T^{2}
61 1+2.07T+61T2 1 + 2.07T + 61T^{2}
67 1−5.73T+67T2 1 - 5.73T + 67T^{2}
71 1+1.12T+71T2 1 + 1.12T + 71T^{2}
73 1+2.99T+73T2 1 + 2.99T + 73T^{2}
79 1−0.409T+79T2 1 - 0.409T + 79T^{2}
83 1+6.57T+83T2 1 + 6.57T + 83T^{2}
89 1+5.62T+89T2 1 + 5.62T + 89T^{2}
97 1+15.4T+97T2 1 + 15.4T + 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

−7.15199572042873426923719030765, −6.71692375353477261524836179701, −5.83185511581699808832932621160, −5.27518684066449789248504410538, −4.89817033771722332152238372187, −4.32896574244844023147954780882, −3.25782943229836725220958056894, −2.53863274215267220527749226992, −1.43767362153808180109493377230, 0, 1.43767362153808180109493377230, 2.53863274215267220527749226992, 3.25782943229836725220958056894, 4.32896574244844023147954780882, 4.89817033771722332152238372187, 5.27518684066449789248504410538, 5.83185511581699808832932621160, 6.71692375353477261524836179701, 7.15199572042873426923719030765

Graph of the ZZ-function along the critical line