Properties

Label 2-1104-1.1-c5-0-38
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s + 59.1·5-s + 213.·7-s + 81·9-s − 126.·11-s − 884.·13-s − 532.·15-s − 1.17e3·17-s + 1.86e3·19-s − 1.91e3·21-s − 529·23-s + 369.·25-s − 729·27-s − 6.78e3·29-s + 5.14e3·31-s + 1.13e3·33-s + 1.26e4·35-s + 5.13e3·37-s + 7.96e3·39-s + 1.24e4·41-s − 4.19e3·43-s + 4.78e3·45-s − 2.30e4·47-s + 2.87e4·49-s + 1.06e4·51-s + 2.51e4·53-s − 7.47e3·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.05·5-s + 1.64·7-s + 0.333·9-s − 0.315·11-s − 1.45·13-s − 0.610·15-s − 0.990·17-s + 1.18·19-s − 0.950·21-s − 0.208·23-s + 0.118·25-s − 0.192·27-s − 1.49·29-s + 0.961·31-s + 0.182·33-s + 1.74·35-s + 0.616·37-s + 0.838·39-s + 1.15·41-s − 0.346·43-s + 0.352·45-s − 1.51·47-s + 1.70·49-s + 0.571·51-s + 1.23·53-s − 0.333·55-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: $\chi_{1104} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.619745701\)
\(L(\frac12)\) \(\approx\) \(2.619745701\)
\(L(\frac{7}{2})\) 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 \( 1 - 59.1T + 3.12e3T^{2} \)
7 \( 1 - 213.T + 1.68e4T^{2} \)
11 \( 1 + 126.T + 1.61e5T^{2} \)
13 \( 1 + 884.T + 3.71e5T^{2} \)
17 \( 1 + 1.17e3T + 1.41e6T^{2} \)
19 \( 1 - 1.86e3T + 2.47e6T^{2} \)
29 \( 1 + 6.78e3T + 2.05e7T^{2} \)
31 \( 1 - 5.14e3T + 2.86e7T^{2} \)
37 \( 1 - 5.13e3T + 6.93e7T^{2} \)
41 \( 1 - 1.24e4T + 1.15e8T^{2} \)
43 \( 1 + 4.19e3T + 1.47e8T^{2} \)
47 \( 1 + 2.30e4T + 2.29e8T^{2} \)
53 \( 1 - 2.51e4T + 4.18e8T^{2} \)
59 \( 1 - 3.71e4T + 7.14e8T^{2} \)
61 \( 1 - 2.64e4T + 8.44e8T^{2} \)
67 \( 1 - 5.43e4T + 1.35e9T^{2} \)
71 \( 1 + 3.56e4T + 1.80e9T^{2} \)
73 \( 1 - 3.39e4T + 2.07e9T^{2} \)
79 \( 1 - 7.66e4T + 3.07e9T^{2} \)
83 \( 1 - 9.66e4T + 3.93e9T^{2} \)
89 \( 1 - 3.00e4T + 5.58e9T^{2} \)
97 \( 1 + 1.46e4T + 8.58e9T^{2} \)
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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

−9.376837489818463934051585563115, −8.155780508294512096787089533567, −7.50279976997768361363030323145, −6.58303243157568296666509564246, −5.34728356545657878823304812281, −5.20093306216359464471033720484, −4.17947168033845633895659373356, −2.43246183019761534113611735159, −1.86775682483801940134749855614, −0.71626281863345919185195161957, 0.71626281863345919185195161957, 1.86775682483801940134749855614, 2.43246183019761534113611735159, 4.17947168033845633895659373356, 5.20093306216359464471033720484, 5.34728356545657878823304812281, 6.58303243157568296666509564246, 7.50279976997768361363030323145, 8.155780508294512096787089533567, 9.376837489818463934051585563115

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