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Properties

Label 2-50-1.1-c3-0-4
Degree 22
Conductor 5050
Sign −1-1
Analytic cond. 2.950092.95009
Root an. cond. 1.717581.71758
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 4·4-s + 4·6-s − 26·7-s − 8·8-s − 23·9-s − 28·11-s − 8·12-s − 12·13-s + 52·14-s + 16·16-s + 64·17-s + 46·18-s − 60·19-s + 52·21-s + 56·22-s + 58·23-s + 16·24-s + 24·26-s + 100·27-s − 104·28-s + 90·29-s − 128·31-s − 32·32-s + 56·33-s − 128·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.384·3-s + 1/2·4-s + 0.272·6-s − 1.40·7-s − 0.353·8-s − 0.851·9-s − 0.767·11-s − 0.192·12-s − 0.256·13-s + 0.992·14-s + 1/4·16-s + 0.913·17-s + 0.602·18-s − 0.724·19-s + 0.540·21-s + 0.542·22-s + 0.525·23-s + 0.136·24-s + 0.181·26-s + 0.712·27-s − 0.701·28-s + 0.576·29-s − 0.741·31-s − 0.176·32-s + 0.295·33-s − 0.645·34-s + ⋯

Functional equation

Λ(s)=(50s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(50s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5050    =    2⋅522 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 2.950092.95009
Root analytic conductor: 1.717581.71758
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 50, ( :3/2), −1)(2,\ 50,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad2 1+pT 1 + p T
5 1 1
good3 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
7 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
11 1+28T+p3T2 1 + 28 T + p^{3} T^{2}
13 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
17 1−64T+p3T2 1 - 64 T + p^{3} T^{2}
19 1+60T+p3T2 1 + 60 T + p^{3} T^{2}
23 1−58T+p3T2 1 - 58 T + p^{3} T^{2}
29 1−90T+p3T2 1 - 90 T + p^{3} T^{2}
31 1+128T+p3T2 1 + 128 T + p^{3} T^{2}
37 1+236T+p3T2 1 + 236 T + p^{3} T^{2}
41 1−242T+p3T2 1 - 242 T + p^{3} T^{2}
43 1+362T+p3T2 1 + 362 T + p^{3} T^{2}
47 1+226T+p3T2 1 + 226 T + p^{3} T^{2}
53 1−108T+p3T2 1 - 108 T + p^{3} T^{2}
59 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
61 1−542T+p3T2 1 - 542 T + p^{3} T^{2}
67 1−434T+p3T2 1 - 434 T + p^{3} T^{2}
71 1+1128T+p3T2 1 + 1128 T + p^{3} T^{2}
73 1+632T+p3T2 1 + 632 T + p^{3} T^{2}
79 1+720T+p3T2 1 + 720 T + p^{3} T^{2}
83 1−478T+p3T2 1 - 478 T + p^{3} T^{2}
89 1+490T+p3T2 1 + 490 T + p^{3} T^{2}
97 1+1456T+p3T2 1 + 1456 T + p^{3} 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

−14.68885980153862411665148753167, −13.12098673613847972078502918692, −12.09285017723653204901966267759, −10.72917816735290999485752832344, −9.753875940004547873505372131129, −8.449707716719759775880307510158, −6.89493100292272320006195407208, −5.62408358393626121672010056962, −3.00920048336052464792456495092, 0, 3.00920048336052464792456495092, 5.62408358393626121672010056962, 6.89493100292272320006195407208, 8.449707716719759775880307510158, 9.753875940004547873505372131129, 10.72917816735290999485752832344, 12.09285017723653204901966267759, 13.12098673613847972078502918692, 14.68885980153862411665148753167

Graph of the ZZ-function along the critical line