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Properties

Label 2-4275-1.1-c1-0-114
Degree 22
Conductor 42754275
Sign −1-1
Analytic cond. 34.136034.1360
Root an. cond. 5.842605.84260
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·7-s + 3·8-s + 6·11-s − 2·14-s − 16-s − 6·17-s + 19-s − 6·22-s − 8·23-s − 2·28-s − 4·29-s − 5·32-s + 6·34-s − 4·37-s − 38-s + 2·43-s − 6·44-s + 8·46-s − 8·47-s − 3·49-s + 2·53-s + 6·56-s + 4·58-s − 12·59-s + 2·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s + 1.80·11-s − 0.534·14-s − 1/4·16-s − 1.45·17-s + 0.229·19-s − 1.27·22-s − 1.66·23-s − 0.377·28-s − 0.742·29-s − 0.883·32-s + 1.02·34-s − 0.657·37-s − 0.162·38-s + 0.304·43-s − 0.904·44-s + 1.17·46-s − 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.801·56-s + 0.525·58-s − 1.56·59-s + 0.256·61-s + ⋯

Functional equation

Λ(s)=(4275s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4275s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 42754275    =    32⋅52⋅193^{2} \cdot 5^{2} \cdot 19
Sign: −1-1
Analytic conductor: 34.136034.1360
Root analytic conductor: 5.842605.84260
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4275, ( :1/2), −1)(2,\ 4275,\ (\ :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
19 1−T 1 - T
good2 1+T+pT2 1 + T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+16T+pT2 1 + 16 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−12T+pT2 1 - 12 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

−8.154904276278665956764754494175, −7.49006725304513024040398275037, −6.65598056508002457182870521609, −5.94645344090271988207277046134, −4.82592459399091433680554684550, −4.27331149587142879061517048995, −3.60839819994467751886447584009, −1.98429554072259245310201630187, −1.40340994003836782410675250116, 0, 1.40340994003836782410675250116, 1.98429554072259245310201630187, 3.60839819994467751886447584009, 4.27331149587142879061517048995, 4.82592459399091433680554684550, 5.94645344090271988207277046134, 6.65598056508002457182870521609, 7.49006725304513024040398275037, 8.154904276278665956764754494175

Graph of the ZZ-function along the critical line