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Properties

Label 2-87e2-1.1-c1-0-65
Degree 22
Conductor 75697569
Sign −1-1
Analytic cond. 60.438760.4387
Root an. cond. 7.774237.77423
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  − 0.854·2-s − 1.26·4-s − 3.95·5-s − 4.38·7-s + 2.79·8-s + 3.38·10-s − 2.89·11-s − 2.95·13-s + 3.74·14-s + 0.149·16-s − 3.86·17-s − 2.14·19-s + 5.02·20-s + 2.47·22-s + 0.0986·23-s + 10.6·25-s + 2.52·26-s + 5.56·28-s − 0.0777·31-s − 5.71·32-s + 3.30·34-s + 17.3·35-s − 10.4·37-s + 1.83·38-s − 11.0·40-s + 6.48·41-s + 1.79·43-s + ⋯
L(s)  = 1  − 0.604·2-s − 0.634·4-s − 1.77·5-s − 1.65·7-s + 0.988·8-s + 1.07·10-s − 0.872·11-s − 0.818·13-s + 1.00·14-s + 0.0373·16-s − 0.936·17-s − 0.492·19-s + 1.12·20-s + 0.527·22-s + 0.0205·23-s + 2.13·25-s + 0.494·26-s + 1.05·28-s − 0.0139·31-s − 1.01·32-s + 0.565·34-s + 2.93·35-s − 1.71·37-s + 0.297·38-s − 1.74·40-s + 1.01·41-s + 0.273·43-s + ⋯

Functional equation

Λ(s)=(7569s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7569s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 75697569    =    32⋅2923^{2} \cdot 29^{2}
Sign: −1-1
Analytic conductor: 60.438760.4387
Root analytic conductor: 7.774237.77423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7569, ( :1/2), −1)(2,\ 7569,\ (\ :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
29 1 1
good2 1+0.854T+2T2 1 + 0.854T + 2T^{2}
5 1+3.95T+5T2 1 + 3.95T + 5T^{2}
7 1+4.38T+7T2 1 + 4.38T + 7T^{2}
11 1+2.89T+11T2 1 + 2.89T + 11T^{2}
13 1+2.95T+13T2 1 + 2.95T + 13T^{2}
17 1+3.86T+17T2 1 + 3.86T + 17T^{2}
19 1+2.14T+19T2 1 + 2.14T + 19T^{2}
23 1−0.0986T+23T2 1 - 0.0986T + 23T^{2}
31 1+0.0777T+31T2 1 + 0.0777T + 31T^{2}
37 1+10.4T+37T2 1 + 10.4T + 37T^{2}
41 1−6.48T+41T2 1 - 6.48T + 41T^{2}
43 1−1.79T+43T2 1 - 1.79T + 43T^{2}
47 1−9.87T+47T2 1 - 9.87T + 47T^{2}
53 1+7.41T+53T2 1 + 7.41T + 53T^{2}
59 1+2.09T+59T2 1 + 2.09T + 59T^{2}
61 1−3.55T+61T2 1 - 3.55T + 61T^{2}
67 1−4.65T+67T2 1 - 4.65T + 67T^{2}
71 1−10.2T+71T2 1 - 10.2T + 71T^{2}
73 1+6.78T+73T2 1 + 6.78T + 73T^{2}
79 1+6.58T+79T2 1 + 6.58T + 79T^{2}
83 1−15.3T+83T2 1 - 15.3T + 83T^{2}
89 1+12.9T+89T2 1 + 12.9T + 89T^{2}
97 1−12.6T+97T2 1 - 12.6T + 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

−7.50791015630819770428916962727, −7.17247853755545180264264458722, −6.38347819258143749160757259354, −5.27780402597005243461734378497, −4.53680879540487061411957687273, −3.91540843483593394598176749019, −3.26169875184758066404336554848, −2.37038883977860532613113038522, −0.57249658571566456496413722985, 0, 0.57249658571566456496413722985, 2.37038883977860532613113038522, 3.26169875184758066404336554848, 3.91540843483593394598176749019, 4.53680879540487061411957687273, 5.27780402597005243461734378497, 6.38347819258143749160757259354, 7.17247853755545180264264458722, 7.50791015630819770428916962727

Graph of the ZZ-function along the critical line