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Properties

Label 2-6525-1.1-c1-0-115
Degree 22
Conductor 65256525
Sign −1-1
Analytic cond. 52.102352.1023
Root an. cond. 7.218197.21819
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  − 1.88·2-s + 1.55·4-s − 1.70·7-s + 0.843·8-s + 2.85·11-s − 6.22·13-s + 3.22·14-s − 4.69·16-s − 1.40·17-s + 6.74·19-s − 5.37·22-s + 1.42·23-s + 11.7·26-s − 2.65·28-s − 29-s − 1.29·31-s + 7.16·32-s + 2.64·34-s + 0.989·37-s − 12.7·38-s − 6.32·41-s − 6.03·43-s + 4.42·44-s − 2.67·46-s + 8.97·47-s − 4.07·49-s − 9.65·52-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.776·4-s − 0.645·7-s + 0.298·8-s + 0.859·11-s − 1.72·13-s + 0.860·14-s − 1.17·16-s − 0.340·17-s + 1.54·19-s − 1.14·22-s + 0.296·23-s + 2.29·26-s − 0.501·28-s − 0.185·29-s − 0.232·31-s + 1.26·32-s + 0.453·34-s + 0.162·37-s − 2.06·38-s − 0.987·41-s − 0.920·43-s + 0.667·44-s − 0.395·46-s + 1.30·47-s − 0.582·49-s − 1.33·52-s + ⋯

Functional equation

Λ(s)=(6525s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6525s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 65256525    =    32⋅52⋅293^{2} \cdot 5^{2} \cdot 29
Sign: −1-1
Analytic conductor: 52.102352.1023
Root analytic conductor: 7.218197.21819
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6525, ( :1/2), −1)(2,\ 6525,\ (\ :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
5 1 1
29 1+T 1 + T
good2 1+1.88T+2T2 1 + 1.88T + 2T^{2}
7 1+1.70T+7T2 1 + 1.70T + 7T^{2}
11 1−2.85T+11T2 1 - 2.85T + 11T^{2}
13 1+6.22T+13T2 1 + 6.22T + 13T^{2}
17 1+1.40T+17T2 1 + 1.40T + 17T^{2}
19 1−6.74T+19T2 1 - 6.74T + 19T^{2}
23 1−1.42T+23T2 1 - 1.42T + 23T^{2}
31 1+1.29T+31T2 1 + 1.29T + 31T^{2}
37 1−0.989T+37T2 1 - 0.989T + 37T^{2}
41 1+6.32T+41T2 1 + 6.32T + 41T^{2}
43 1+6.03T+43T2 1 + 6.03T + 43T^{2}
47 1−8.97T+47T2 1 - 8.97T + 47T^{2}
53 1−2.11T+53T2 1 - 2.11T + 53T^{2}
59 1−10.0T+59T2 1 - 10.0T + 59T^{2}
61 1−5.73T+61T2 1 - 5.73T + 61T^{2}
67 1−2.45T+67T2 1 - 2.45T + 67T^{2}
71 1+5.04T+71T2 1 + 5.04T + 71T^{2}
73 1+5.55T+73T2 1 + 5.55T + 73T^{2}
79 1−9.37T+79T2 1 - 9.37T + 79T^{2}
83 1−5.06T+83T2 1 - 5.06T + 83T^{2}
89 1−1.75T+89T2 1 - 1.75T + 89T^{2}
97 1+14.6T+97T2 1 + 14.6T + 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

−7.71156608761787966663473671349, −6.95773189249016534951860335205, −6.81653080654946286041306408276, −5.54042382728503136804026488468, −4.86604047966318953919667409167, −3.92913199049652372904232089117, −2.95954807655101570700128708896, −2.07063000141996437858675998767, −1.03860224928909594135275134159, 0, 1.03860224928909594135275134159, 2.07063000141996437858675998767, 2.95954807655101570700128708896, 3.92913199049652372904232089117, 4.86604047966318953919667409167, 5.54042382728503136804026488468, 6.81653080654946286041306408276, 6.95773189249016534951860335205, 7.71156608761787966663473671349

Graph of the ZZ-function along the critical line