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Properties

Label 2-23275-1.1-c1-0-16
Degree 22
Conductor 2327523275
Sign −1-1
Analytic cond. 185.851185.851
Root an. cond. 13.632713.6327
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 + 3·8-s − 3·9-s − 4·11-s + 2·13-s − 16-s − 4·17-s + 3·18-s − 19-s + 4·22-s − 6·23-s − 2·26-s − 6·29-s + 4·31-s − 5·32-s + 4·34-s + 3·36-s − 10·37-s + 38-s + 10·41-s + 2·43-s + 4·44-s + 6·46-s + 6·47-s − 2·52-s + 10·53-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s − 9-s − 1.20·11-s + 0.554·13-s − 1/4·16-s − 0.970·17-s + 0.707·18-s − 0.229·19-s + 0.852·22-s − 1.25·23-s − 0.392·26-s − 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.685·34-s + 1/2·36-s − 1.64·37-s + 0.162·38-s + 1.56·41-s + 0.304·43-s + 0.603·44-s + 0.884·46-s + 0.875·47-s − 0.277·52-s + 1.37·53-s + ⋯

Functional equation

Λ(s)=(23275s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 23275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(23275s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 23275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2327523275    =    52⋅72⋅195^{2} \cdot 7^{2} \cdot 19
Sign: −1-1
Analytic conductor: 185.851185.851
Root analytic conductor: 13.632713.6327
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 23275, ( :1/2), −1)(2,\ 23275,\ (\ :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)
bad5 1 1
7 1 1
19 1+T 1 + T
good2 1+T+pT2 1 + T + p T^{2}
3 1+pT2 1 + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−18T+pT2 1 - 18 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1+6T+pT2 1 + 6 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

−15.80927037546874, −15.34469056627237, −14.59920217656141, −14.00353056475203, −13.55577503339657, −13.23198965962026, −12.47137782871371, −11.92897050442017, −11.09315478279226, −10.72903292838708, −10.35626718664521, −9.513586823604723, −9.100133175352587, −8.438119017371597, −8.160651458844081, −7.540779182409274, −6.853301141604641, −5.931613350329385, −5.558072484862187, −4.840219553666075, −4.106521831421743, −3.512497882161801, −2.451671303288168, −2.002941568472231, −0.7287804435836824, 0, 0.7287804435836824, 2.002941568472231, 2.451671303288168, 3.512497882161801, 4.106521831421743, 4.840219553666075, 5.558072484862187, 5.931613350329385, 6.853301141604641, 7.540779182409274, 8.160651458844081, 8.438119017371597, 9.100133175352587, 9.513586823604723, 10.35626718664521, 10.72903292838708, 11.09315478279226, 11.92897050442017, 12.47137782871371, 13.23198965962026, 13.55577503339657, 14.00353056475203, 14.59920217656141, 15.34469056627237, 15.80927037546874

Graph of the ZZ-function along the critical line