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Properties

Label 2-7800-1.1-c1-0-96
Degree 22
Conductor 78007800
Sign −1-1
Analytic cond. 62.283362.2833
Root an. cond. 7.891977.89197
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  + 3-s + 9-s − 2.86·11-s + 13-s − 5.52·17-s + 3.52·19-s − 7.52·23-s + 27-s + 6.77·29-s + 5.72·31-s − 2.86·33-s + 3.72·37-s + 39-s − 10.1·41-s + 5.52·43-s − 8.65·47-s − 7·49-s − 5.52·51-s + 6.77·53-s + 3.52·57-s + 0.593·59-s − 5.25·61-s + 10.5·67-s − 7.52·69-s − 2.38·71-s − 5.45·73-s + 2.47·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.333·9-s − 0.863·11-s + 0.277·13-s − 1.33·17-s + 0.808·19-s − 1.56·23-s + 0.192·27-s + 1.25·29-s + 1.02·31-s − 0.498·33-s + 0.613·37-s + 0.160·39-s − 1.58·41-s + 0.842·43-s − 1.26·47-s − 49-s − 0.773·51-s + 0.930·53-s + 0.466·57-s + 0.0773·59-s − 0.672·61-s + 1.28·67-s − 0.905·69-s − 0.283·71-s − 0.638·73-s + 0.278·79-s + ⋯

Functional equation

Λ(s)=(7800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 78007800    =    23⋅3⋅52⋅132^{3} \cdot 3 \cdot 5^{2} \cdot 13
Sign: −1-1
Analytic conductor: 62.283362.2833
Root analytic conductor: 7.891977.89197
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7800, ( :1/2), −1)(2,\ 7800,\ (\ :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)
bad2 1 1
3 1−T 1 - T
5 1 1
13 1−T 1 - T
good7 1+7T2 1 + 7T^{2}
11 1+2.86T+11T2 1 + 2.86T + 11T^{2}
17 1+5.52T+17T2 1 + 5.52T + 17T^{2}
19 1−3.52T+19T2 1 - 3.52T + 19T^{2}
23 1+7.52T+23T2 1 + 7.52T + 23T^{2}
29 1−6.77T+29T2 1 - 6.77T + 29T^{2}
31 1−5.72T+31T2 1 - 5.72T + 31T^{2}
37 1−3.72T+37T2 1 - 3.72T + 37T^{2}
41 1+10.1T+41T2 1 + 10.1T + 41T^{2}
43 1−5.52T+43T2 1 - 5.52T + 43T^{2}
47 1+8.65T+47T2 1 + 8.65T + 47T^{2}
53 1−6.77T+53T2 1 - 6.77T + 53T^{2}
59 1−0.593T+59T2 1 - 0.593T + 59T^{2}
61 1+5.25T+61T2 1 + 5.25T + 61T^{2}
67 1−10.5T+67T2 1 - 10.5T + 67T^{2}
71 1+2.38T+71T2 1 + 2.38T + 71T^{2}
73 1+5.45T+73T2 1 + 5.45T + 73T^{2}
79 1−2.47T+79T2 1 - 2.47T + 79T^{2}
83 1+8.11T+83T2 1 + 8.11T + 83T^{2}
89 1+14.1T+89T2 1 + 14.1T + 89T^{2}
97 1+6T+97T2 1 + 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

−7.67215588937917696884675765736, −6.77953939702828842126594776782, −6.27699128499688535294274464680, −5.33471249725517386297250747019, −4.60479022217698224353097100293, −3.94130844640860387594086373671, −2.95804649553138258896601839949, −2.39642490723420324740181388671, −1.38887103708305694508247303011, 0, 1.38887103708305694508247303011, 2.39642490723420324740181388671, 2.95804649553138258896601839949, 3.94130844640860387594086373671, 4.60479022217698224353097100293, 5.33471249725517386297250747019, 6.27699128499688535294274464680, 6.77953939702828842126594776782, 7.67215588937917696884675765736

Graph of the ZZ-function along the critical line