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Properties

Label 2-6525-1.1-c1-0-126
Degree 22
Conductor 65256525
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·2-s − 1.61·4-s + 3·7-s − 2.23·8-s + 5.47·11-s + 6.23·13-s + 1.85·14-s + 1.85·16-s + 3.47·17-s + 7.70·19-s + 3.38·22-s + 3.85·26-s − 4.85·28-s − 29-s − 8·31-s + 5.61·32-s + 2.14·34-s + 8·37-s + 4.76·38-s + 4.47·41-s − 3.23·43-s − 8.85·44-s + 6.70·47-s + 2·49-s − 10.0·52-s − 6.76·53-s − 6.70·56-s + ⋯
L(s)  = 1  + 0.437·2-s − 0.809·4-s + 1.13·7-s − 0.790·8-s + 1.64·11-s + 1.72·13-s + 0.495·14-s + 0.463·16-s + 0.842·17-s + 1.76·19-s + 0.721·22-s + 0.755·26-s − 0.917·28-s − 0.185·29-s − 1.43·31-s + 0.993·32-s + 0.368·34-s + 1.31·37-s + 0.772·38-s + 0.698·41-s − 0.493·43-s − 1.33·44-s + 0.978·47-s + 0.285·49-s − 1.39·52-s − 0.929·53-s − 0.896·56-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: 11
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: 00
Selberg data: (2, 6525, ( :1/2), 1)(2,\ 6525,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2646059183.264605918
L(12)L(\frac12) ≈\approx 3.2646059183.264605918
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 1−0.618T+2T2 1 - 0.618T + 2T^{2}
7 1−3T+7T2 1 - 3T + 7T^{2}
11 1−5.47T+11T2 1 - 5.47T + 11T^{2}
13 1−6.23T+13T2 1 - 6.23T + 13T^{2}
17 1−3.47T+17T2 1 - 3.47T + 17T^{2}
19 1−7.70T+19T2 1 - 7.70T + 19T^{2}
23 1+23T2 1 + 23T^{2}
31 1+8T+31T2 1 + 8T + 31T^{2}
37 1−8T+37T2 1 - 8T + 37T^{2}
41 1−4.47T+41T2 1 - 4.47T + 41T^{2}
43 1+3.23T+43T2 1 + 3.23T + 43T^{2}
47 1−6.70T+47T2 1 - 6.70T + 47T^{2}
53 1+6.76T+53T2 1 + 6.76T + 53T^{2}
59 1+5.23T+59T2 1 + 5.23T + 59T^{2}
61 1+5.70T+61T2 1 + 5.70T + 61T^{2}
67 1−11.4T+67T2 1 - 11.4T + 67T^{2}
71 1+7.23T+71T2 1 + 7.23T + 71T^{2}
73 1−8T+73T2 1 - 8T + 73T^{2}
79 1+6.18T+79T2 1 + 6.18T + 79T^{2}
83 1+3.70T+83T2 1 + 3.70T + 83T^{2}
89 1+11.1T+89T2 1 + 11.1T + 89T^{2}
97 1+2.76T+97T2 1 + 2.76T + 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

−8.040406429657753405828417545233, −7.43527198631723347953430121433, −6.38422253454370398577104981968, −5.72899459123284017770723157425, −5.21770040889114517555638204420, −4.23795454770776330975962457280, −3.80058502672385595447266020499, −3.10361143848694433688060143986, −1.43888579542215107097389660009, −1.08636406122801098652270404839, 1.08636406122801098652270404839, 1.43888579542215107097389660009, 3.10361143848694433688060143986, 3.80058502672385595447266020499, 4.23795454770776330975962457280, 5.21770040889114517555638204420, 5.72899459123284017770723157425, 6.38422253454370398577104981968, 7.43527198631723347953430121433, 8.040406429657753405828417545233

Graph of the ZZ-function along the critical line