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Properties

Label 2-2400-1.1-c1-0-36
Degree 22
Conductor 24002400
Sign −1-1
Analytic cond. 19.164019.1640
Root an. cond. 4.377684.37768
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  + 3-s + 9-s − 2·13-s − 6·17-s − 4·19-s − 8·23-s + 27-s − 2·29-s + 4·31-s − 10·37-s − 2·39-s + 2·41-s + 4·43-s − 8·47-s − 7·49-s − 6·51-s + 2·53-s − 4·57-s + 8·59-s − 2·61-s + 12·67-s − 8·69-s + 8·71-s + 14·73-s − 12·79-s + 81-s + 4·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.554·13-s − 1.45·17-s − 0.917·19-s − 1.66·23-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 1.64·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s − 1.16·47-s − 49-s − 0.840·51-s + 0.274·53-s − 0.529·57-s + 1.04·59-s − 0.256·61-s + 1.46·67-s − 0.963·69-s + 0.949·71-s + 1.63·73-s − 1.35·79-s + 1/9·81-s + 0.439·83-s + ⋯

Functional equation

Λ(s)=(2400s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2400s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 19.164019.1640
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2400, ( :1/2), −1)(2,\ 2400,\ (\ :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)
bad2 1 1
3 1−T 1 - T
5 1 1
good7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+2T+pT2 1 + 2 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 1−2T+pT2 1 - 2 T + p T^{2}
43 1−4T+pT2 1 - 4 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 1−8T+pT2 1 - 8 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−8.403731101485364375376192345699, −8.090705707741763680917159262620, −6.93734943901991982578238677077, −6.47967036650562281115929462317, −5.36153901167379763944515499710, −4.41269930710391273354556396845, −3.76487004494944959266400194993, −2.52766240342707528511307153079, −1.85990170159571189913185703867, 0, 1.85990170159571189913185703867, 2.52766240342707528511307153079, 3.76487004494944959266400194993, 4.41269930710391273354556396845, 5.36153901167379763944515499710, 6.47967036650562281115929462317, 6.93734943901991982578238677077, 8.090705707741763680917159262620, 8.403731101485364375376192345699

Graph of the ZZ-function along the critical line