Properties

Label 2-1104-1.1-c1-0-2
Degree $2$
Conductor $1104$
Sign $1$
Analytic cond. $8.81548$
Root an. cond. $2.96908$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.16·5-s − 5.16·7-s + 9-s + 4·13-s − 3.16·15-s + 7.16·17-s + 1.16·19-s − 5.16·21-s + 23-s + 5.00·25-s + 27-s + 8.32·29-s + 6.32·31-s + 16.3·35-s − 8.32·37-s + 4·39-s − 2·41-s − 1.16·43-s − 3.16·45-s − 0.324·47-s + 19.6·49-s + 7.16·51-s + 5.48·53-s + 1.16·57-s + 8.32·59-s + 0.324·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.41·5-s − 1.95·7-s + 0.333·9-s + 1.10·13-s − 0.816·15-s + 1.73·17-s + 0.266·19-s − 1.12·21-s + 0.208·23-s + 1.00·25-s + 0.192·27-s + 1.54·29-s + 1.13·31-s + 2.75·35-s − 1.36·37-s + 0.640·39-s − 0.312·41-s − 0.177·43-s − 0.471·45-s − 0.0473·47-s + 2.80·49-s + 1.00·51-s + 0.753·53-s + 0.153·57-s + 1.08·59-s + 0.0415·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $1$
Analytic conductor: \(8.81548\)
Root analytic conductor: \(2.96908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.255537043\)
\(L(\frac12)\) \(\approx\) \(1.255537043\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 - T \)
good5 \( 1 + 3.16T + 5T^{2} \)
7 \( 1 + 5.16T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 7.16T + 17T^{2} \)
19 \( 1 - 1.16T + 19T^{2} \)
29 \( 1 - 8.32T + 29T^{2} \)
31 \( 1 - 6.32T + 31T^{2} \)
37 \( 1 + 8.32T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 1.16T + 43T^{2} \)
47 \( 1 + 0.324T + 47T^{2} \)
53 \( 1 - 5.48T + 53T^{2} \)
59 \( 1 - 8.32T + 59T^{2} \)
61 \( 1 - 0.324T + 61T^{2} \)
67 \( 1 + 1.16T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 + 3.67T + 97T^{2} \)
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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

−10.03076173396585605378171377304, −8.837489977467702222922574464784, −8.321446872170161105388013620011, −7.35106672945793279679124362572, −6.68868166127075242542647762744, −5.72431909816859384934912545255, −4.26572756644442675097690500642, −3.33761895695322852109252170446, −3.11356927160187334654369731050, −0.833170383623590513029395625074, 0.833170383623590513029395625074, 3.11356927160187334654369731050, 3.33761895695322852109252170446, 4.26572756644442675097690500642, 5.72431909816859384934912545255, 6.68868166127075242542647762744, 7.35106672945793279679124362572, 8.321446872170161105388013620011, 8.837489977467702222922574464784, 10.03076173396585605378171377304

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