Properties

Label 2-1104-1.1-c5-0-108
Degree $2$
Conductor $1104$
Sign $-1$
Analytic cond. $177.063$
Root an. cond. $13.3065$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s + 19.9·5-s + 195.·7-s + 81·9-s − 378.·11-s + 679.·13-s + 179.·15-s − 732.·17-s − 1.12e3·19-s + 1.75e3·21-s − 529·23-s − 2.72e3·25-s + 729·27-s − 6.56e3·29-s − 5.42e3·31-s − 3.40e3·33-s + 3.90e3·35-s + 2.00e3·37-s + 6.11e3·39-s − 1.14e4·41-s − 6.40e3·43-s + 1.61e3·45-s − 2.56e4·47-s + 2.12e4·49-s − 6.59e3·51-s − 2.35e4·53-s − 7.56e3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.357·5-s + 1.50·7-s + 0.333·9-s − 0.943·11-s + 1.11·13-s + 0.206·15-s − 0.614·17-s − 0.712·19-s + 0.868·21-s − 0.208·23-s − 0.872·25-s + 0.192·27-s − 1.45·29-s − 1.01·31-s − 0.544·33-s + 0.538·35-s + 0.240·37-s + 0.644·39-s − 1.05·41-s − 0.528·43-s + 0.119·45-s − 1.69·47-s + 1.26·49-s − 0.354·51-s − 1.15·53-s − 0.337·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $-1$
Analytic conductor: \(177.063\)
Root analytic conductor: \(13.3065\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1104,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 9T \)
23 \( 1 + 529T \)
good5 \( 1 - 19.9T + 3.12e3T^{2} \)
7 \( 1 - 195.T + 1.68e4T^{2} \)
11 \( 1 + 378.T + 1.61e5T^{2} \)
13 \( 1 - 679.T + 3.71e5T^{2} \)
17 \( 1 + 732.T + 1.41e6T^{2} \)
19 \( 1 + 1.12e3T + 2.47e6T^{2} \)
29 \( 1 + 6.56e3T + 2.05e7T^{2} \)
31 \( 1 + 5.42e3T + 2.86e7T^{2} \)
37 \( 1 - 2.00e3T + 6.93e7T^{2} \)
41 \( 1 + 1.14e4T + 1.15e8T^{2} \)
43 \( 1 + 6.40e3T + 1.47e8T^{2} \)
47 \( 1 + 2.56e4T + 2.29e8T^{2} \)
53 \( 1 + 2.35e4T + 4.18e8T^{2} \)
59 \( 1 - 3.84e4T + 7.14e8T^{2} \)
61 \( 1 + 5.27e4T + 8.44e8T^{2} \)
67 \( 1 - 6.54e4T + 1.35e9T^{2} \)
71 \( 1 + 3.28e4T + 1.80e9T^{2} \)
73 \( 1 + 8.84e4T + 2.07e9T^{2} \)
79 \( 1 - 3.41e4T + 3.07e9T^{2} \)
83 \( 1 + 8.94e3T + 3.93e9T^{2} \)
89 \( 1 + 1.02e5T + 5.58e9T^{2} \)
97 \( 1 - 1.75e5T + 8.58e9T^{2} \)
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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

−8.494914119519387017692258781806, −8.107229408678307476719615279204, −7.25972569251645180892016847824, −6.09109458336394821782870000179, −5.23151804280840134216374684671, −4.36704603766844641933195361748, −3.38006370019043877858152117481, −2.00538373705566715628974449789, −1.65163876727299795939449135605, 0, 1.65163876727299795939449135605, 2.00538373705566715628974449789, 3.38006370019043877858152117481, 4.36704603766844641933195361748, 5.23151804280840134216374684671, 6.09109458336394821782870000179, 7.25972569251645180892016847824, 8.107229408678307476719615279204, 8.494914119519387017692258781806

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