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Properties

Label 2-6525-1.1-c1-0-107
Degree 22
Conductor 65256525
Sign 11
Analytic cond. 52.102352.1023
Root an. cond. 7.218197.21819
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 2·7-s + 3·11-s + 4·13-s + 4·14-s − 4·16-s − 2·17-s − 2·19-s + 6·22-s + 5·23-s + 8·26-s + 4·28-s + 29-s + 2·31-s − 8·32-s − 4·34-s + 5·37-s − 4·38-s + 41-s + 43-s + 6·44-s + 10·46-s + 6·47-s − 3·49-s + 8·52-s + 3·53-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.755·7-s + 0.904·11-s + 1.10·13-s + 1.06·14-s − 16-s − 0.485·17-s − 0.458·19-s + 1.27·22-s + 1.04·23-s + 1.56·26-s + 0.755·28-s + 0.185·29-s + 0.359·31-s − 1.41·32-s − 0.685·34-s + 0.821·37-s − 0.648·38-s + 0.156·41-s + 0.152·43-s + 0.904·44-s + 1.47·46-s + 0.875·47-s − 3/7·49-s + 1.10·52-s + 0.412·53-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: yes
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 5.3064765995.306476599
L(12)L(\frac12) ≈\approx 5.3064765995.306476599
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−pT+pT2 1 - p T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−5T+pT2 1 - 5 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−5T+pT2 1 - 5 T + p T^{2}
41 1−T+pT2 1 - T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−3T+pT2 1 - 3 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+9T+pT2 1 + 9 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1−7T+pT2 1 - 7 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−13T+pT2 1 - 13 T + p T^{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

−7.918311564552459896414129871883, −6.99219953777008933414075068338, −6.37793889190286755907620802267, −5.84982517389395981659595838466, −4.99342273522673413054819938011, −4.38244342721143014380994810645, −3.83671545156978843468561546559, −3.00570317852034526630587963516, −2.06295820671583417203795756531, −1.01836719130596143283894004150, 1.01836719130596143283894004150, 2.06295820671583417203795756531, 3.00570317852034526630587963516, 3.83671545156978843468561546559, 4.38244342721143014380994810645, 4.99342273522673413054819938011, 5.84982517389395981659595838466, 6.37793889190286755907620802267, 6.99219953777008933414075068338, 7.918311564552459896414129871883

Graph of the ZZ-function along the critical line