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Properties

Label 2-8800-1.1-c1-0-124
Degree $2$
Conductor $8800$
Sign $-1$
Analytic cond. $70.2683$
Root an. cond. $8.38262$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 3·7-s + 6·9-s − 11-s − 4·13-s + 7·17-s − 3·19-s − 9·21-s − 9·27-s + 7·29-s + 3·31-s + 3·33-s − 9·37-s + 12·39-s − 4·41-s − 6·43-s − 12·47-s + 2·49-s − 21·51-s + 7·53-s + 9·57-s + 12·59-s + 61-s + 18·63-s − 12·67-s + 9·71-s + 2·73-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.13·7-s + 2·9-s − 0.301·11-s − 1.10·13-s + 1.69·17-s − 0.688·19-s − 1.96·21-s − 1.73·27-s + 1.29·29-s + 0.538·31-s + 0.522·33-s − 1.47·37-s + 1.92·39-s − 0.624·41-s − 0.914·43-s − 1.75·47-s + 2/7·49-s − 2.94·51-s + 0.961·53-s + 1.19·57-s + 1.56·59-s + 0.128·61-s + 2.26·63-s − 1.46·67-s + 1.06·71-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8800\)    =    \(2^{5} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(70.2683\)
Root analytic conductor: \(8.38262\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8800,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.16563472775969471348455457448, −6.77709962632504940122183775431, −5.87317627869163144373234120959, −5.18367221616459560452710696971, −4.96846666168592226363450444810, −4.23231658118273547411260964461, −3.08569747085328047651116679020, −1.87511194260782983356744326004, −1.09596217990125069100252676038, 0, 1.09596217990125069100252676038, 1.87511194260782983356744326004, 3.08569747085328047651116679020, 4.23231658118273547411260964461, 4.96846666168592226363450444810, 5.18367221616459560452710696971, 5.87317627869163144373234120959, 6.77709962632504940122183775431, 7.16563472775969471348455457448

Graph of the $Z$-function along the critical line