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Properties

Label 2-6525-1.1-c1-0-106
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  − 1.23·2-s − 0.471·4-s − 3.27·7-s + 3.05·8-s − 2.30·11-s + 5.57·13-s + 4.05·14-s − 2.83·16-s + 1.94·17-s − 3.59·19-s + 2.84·22-s + 1.66·23-s − 6.89·26-s + 1.54·28-s − 29-s + 1.70·31-s − 2.60·32-s − 2.39·34-s − 9.16·37-s + 4.44·38-s + 8.54·41-s − 3.56·43-s + 1.08·44-s − 2.05·46-s − 11.5·47-s + 3.73·49-s − 2.62·52-s + ⋯
L(s)  = 1  − 0.874·2-s − 0.235·4-s − 1.23·7-s + 1.08·8-s − 0.694·11-s + 1.54·13-s + 1.08·14-s − 0.708·16-s + 0.470·17-s − 0.823·19-s + 0.606·22-s + 0.346·23-s − 1.35·26-s + 0.291·28-s − 0.185·29-s + 0.306·31-s − 0.460·32-s − 0.411·34-s − 1.50·37-s + 0.720·38-s + 1.33·41-s − 0.543·43-s + 0.163·44-s − 0.303·46-s − 1.68·47-s + 0.533·49-s − 0.364·52-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+1.23T+2T2 1 + 1.23T + 2T^{2}
7 1+3.27T+7T2 1 + 3.27T + 7T^{2}
11 1+2.30T+11T2 1 + 2.30T + 11T^{2}
13 1−5.57T+13T2 1 - 5.57T + 13T^{2}
17 1−1.94T+17T2 1 - 1.94T + 17T^{2}
19 1+3.59T+19T2 1 + 3.59T + 19T^{2}
23 1−1.66T+23T2 1 - 1.66T + 23T^{2}
31 1−1.70T+31T2 1 - 1.70T + 31T^{2}
37 1+9.16T+37T2 1 + 9.16T + 37T^{2}
41 1−8.54T+41T2 1 - 8.54T + 41T^{2}
43 1+3.56T+43T2 1 + 3.56T + 43T^{2}
47 1+11.5T+47T2 1 + 11.5T + 47T^{2}
53 1+9.66T+53T2 1 + 9.66T + 53T^{2}
59 1−9.83T+59T2 1 - 9.83T + 59T^{2}
61 1−5.42T+61T2 1 - 5.42T + 61T^{2}
67 1−5.20T+67T2 1 - 5.20T + 67T^{2}
71 1−6.02T+71T2 1 - 6.02T + 71T^{2}
73 1−15.5T+73T2 1 - 15.5T + 73T^{2}
79 1−12.2T+79T2 1 - 12.2T + 79T^{2}
83 1+10.7T+83T2 1 + 10.7T + 83T^{2}
89 1−2.53T+89T2 1 - 2.53T + 89T^{2}
97 1+5.89T+97T2 1 + 5.89T + 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

−8.015148250592183757793907886141, −6.91098101913074315099803991585, −6.47277597439704426092090931430, −5.60728076536332471115212359258, −4.82544504531333511034320019172, −3.78100484164590848274556824765, −3.31576338829723681025399213314, −2.11055156903839115512337914651, −1.01212440022278897712629764799, 0, 1.01212440022278897712629764799, 2.11055156903839115512337914651, 3.31576338829723681025399213314, 3.78100484164590848274556824765, 4.82544504531333511034320019172, 5.60728076536332471115212359258, 6.47277597439704426092090931430, 6.91098101913074315099803991585, 8.015148250592183757793907886141

Graph of the ZZ-function along the critical line