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Properties

Label 2-6525-1.1-c1-0-170
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  − 2.57·2-s + 4.64·4-s + 4.69·7-s − 6.81·8-s + 3.11·11-s − 5.07·13-s − 12.1·14-s + 8.28·16-s − 1.40·17-s − 3.76·19-s − 8.03·22-s − 5.71·23-s + 13.0·26-s + 21.8·28-s − 29-s − 2.23·31-s − 7.73·32-s + 3.60·34-s + 5.79·37-s + 9.70·38-s + 10.6·41-s − 8.89·43-s + 14.4·44-s + 14.7·46-s − 3.62·47-s + 15.0·49-s − 23.5·52-s + ⋯
L(s)  = 1  − 1.82·2-s + 2.32·4-s + 1.77·7-s − 2.41·8-s + 0.939·11-s − 1.40·13-s − 3.23·14-s + 2.07·16-s − 0.339·17-s − 0.863·19-s − 1.71·22-s − 1.19·23-s + 2.56·26-s + 4.12·28-s − 0.185·29-s − 0.400·31-s − 1.36·32-s + 0.619·34-s + 0.952·37-s + 1.57·38-s + 1.67·41-s − 1.35·43-s + 2.18·44-s + 2.17·46-s − 0.529·47-s + 2.15·49-s − 3.27·52-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+2.57T+2T2 1 + 2.57T + 2T^{2}
7 1−4.69T+7T2 1 - 4.69T + 7T^{2}
11 1−3.11T+11T2 1 - 3.11T + 11T^{2}
13 1+5.07T+13T2 1 + 5.07T + 13T^{2}
17 1+1.40T+17T2 1 + 1.40T + 17T^{2}
19 1+3.76T+19T2 1 + 3.76T + 19T^{2}
23 1+5.71T+23T2 1 + 5.71T + 23T^{2}
31 1+2.23T+31T2 1 + 2.23T + 31T^{2}
37 1−5.79T+37T2 1 - 5.79T + 37T^{2}
41 1−10.6T+41T2 1 - 10.6T + 41T^{2}
43 1+8.89T+43T2 1 + 8.89T + 43T^{2}
47 1+3.62T+47T2 1 + 3.62T + 47T^{2}
53 1−0.948T+53T2 1 - 0.948T + 53T^{2}
59 1−8.53T+59T2 1 - 8.53T + 59T^{2}
61 1+6.21T+61T2 1 + 6.21T + 61T^{2}
67 1+13.9T+67T2 1 + 13.9T + 67T^{2}
71 1+5.88T+71T2 1 + 5.88T + 71T^{2}
73 1−7.08T+73T2 1 - 7.08T + 73T^{2}
79 1−7.31T+79T2 1 - 7.31T + 79T^{2}
83 1+13.6T+83T2 1 + 13.6T + 83T^{2}
89 1−7.98T+89T2 1 - 7.98T + 89T^{2}
97 1+13.8T+97T2 1 + 13.8T + 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

−7.78280190628978762509429401210, −7.35485435371872029782794761762, −6.56803133553546558619632274712, −5.78762875277821434528167501092, −4.74665820987326923863898530779, −4.08683926466824348439100413826, −2.55318381383418278076963569558, −1.96613109548948213049500434777, −1.27220129901495762124123306942, 0, 1.27220129901495762124123306942, 1.96613109548948213049500434777, 2.55318381383418278076963569558, 4.08683926466824348439100413826, 4.74665820987326923863898530779, 5.78762875277821434528167501092, 6.56803133553546558619632274712, 7.35485435371872029782794761762, 7.78280190628978762509429401210

Graph of the ZZ-function along the critical line