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Properties

Label 2-2400-1.1-c3-0-94
Degree 22
Conductor 24002400
Sign −1-1
Analytic cond. 141.604141.604
Root an. cond. 11.899711.8997
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  + 3·3-s − 8·7-s + 9·9-s − 4·11-s + 6·13-s + 2·17-s + 16·19-s − 24·21-s − 60·23-s + 27·27-s − 142·29-s + 176·31-s − 12·33-s + 214·37-s + 18·39-s − 278·41-s − 68·43-s + 116·47-s − 279·49-s + 6·51-s + 350·53-s + 48·57-s − 684·59-s − 394·61-s − 72·63-s + 108·67-s − 180·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.431·7-s + 1/3·9-s − 0.109·11-s + 0.128·13-s + 0.0285·17-s + 0.193·19-s − 0.249·21-s − 0.543·23-s + 0.192·27-s − 0.909·29-s + 1.01·31-s − 0.0633·33-s + 0.950·37-s + 0.0739·39-s − 1.05·41-s − 0.241·43-s + 0.360·47-s − 0.813·49-s + 0.0164·51-s + 0.907·53-s + 0.111·57-s − 1.50·59-s − 0.826·61-s − 0.143·63-s + 0.196·67-s − 0.314·69-s + ⋯

Functional equation

Λ(s)=(2400s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2400s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24002400    =    25⋅3⋅522^{5} \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 141.604141.604
Root analytic conductor: 11.899711.8997
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2400, ( :3/2), −1)(2,\ 2400,\ (\ :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 1
3 1−pT 1 - p T
5 1 1
good7 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
11 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
13 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
17 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
19 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
23 1+60T+p3T2 1 + 60 T + p^{3} T^{2}
29 1+142T+p3T2 1 + 142 T + p^{3} T^{2}
31 1−176T+p3T2 1 - 176 T + p^{3} T^{2}
37 1−214T+p3T2 1 - 214 T + p^{3} T^{2}
41 1+278T+p3T2 1 + 278 T + p^{3} T^{2}
43 1+68T+p3T2 1 + 68 T + p^{3} T^{2}
47 1−116T+p3T2 1 - 116 T + p^{3} T^{2}
53 1−350T+p3T2 1 - 350 T + p^{3} T^{2}
59 1+684T+p3T2 1 + 684 T + p^{3} T^{2}
61 1+394T+p3T2 1 + 394 T + p^{3} T^{2}
67 1−108T+p3T2 1 - 108 T + p^{3} T^{2}
71 1−96T+p3T2 1 - 96 T + p^{3} T^{2}
73 1−398T+p3T2 1 - 398 T + p^{3} T^{2}
79 1+136T+p3T2 1 + 136 T + p^{3} T^{2}
83 1−436T+p3T2 1 - 436 T + p^{3} T^{2}
89 1+750T+p3T2 1 + 750 T + p^{3} T^{2}
97 1+82T+p3T2 1 + 82 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

−8.182693680591538666959096617752, −7.60676236115966867735900060715, −6.69316860285617402076303082902, −5.99278568986096849768324695264, −4.99816187385205939099420184439, −4.06896616518560573632059141234, −3.26098754478655631189795926997, −2.40186957324362766207447646118, −1.31761933060326068687825865592, 0, 1.31761933060326068687825865592, 2.40186957324362766207447646118, 3.26098754478655631189795926997, 4.06896616518560573632059141234, 4.99816187385205939099420184439, 5.99278568986096849768324695264, 6.69316860285617402076303082902, 7.60676236115966867735900060715, 8.182693680591538666959096617752

Graph of the ZZ-function along the critical line