Properties

Label 2-1183-1.1-c1-0-45
Degree $2$
Conductor $1183$
Sign $1$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 3·5-s + 7-s − 3·9-s + 6·10-s + 6·11-s + 2·14-s − 4·16-s + 4·17-s − 6·18-s − 5·19-s + 6·20-s + 12·22-s + 3·23-s + 4·25-s + 2·28-s − 5·29-s + 3·31-s − 8·32-s + 8·34-s + 3·35-s − 6·36-s + 4·37-s − 10·38-s + 6·41-s − 43-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.34·5-s + 0.377·7-s − 9-s + 1.89·10-s + 1.80·11-s + 0.534·14-s − 16-s + 0.970·17-s − 1.41·18-s − 1.14·19-s + 1.34·20-s + 2.55·22-s + 0.625·23-s + 4/5·25-s + 0.377·28-s − 0.928·29-s + 0.538·31-s − 1.41·32-s + 1.37·34-s + 0.507·35-s − 36-s + 0.657·37-s − 1.62·38-s + 0.937·41-s − 0.152·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.323622813\)
\(L(\frac12)\) \(\approx\) \(4.323622813\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−9.536451784505074313808134105535, −9.207834379220788354016903344416, −8.157183950347712692726002057248, −6.69002626594602126625681974705, −6.16699839705098439022921833193, −5.57652565474586077264622741418, −4.66814317251753947271974703426, −3.69113279097650649608530641419, −2.68417503048537603951806988505, −1.57613473105511884594471340976, 1.57613473105511884594471340976, 2.68417503048537603951806988505, 3.69113279097650649608530641419, 4.66814317251753947271974703426, 5.57652565474586077264622741418, 6.16699839705098439022921833193, 6.69002626594602126625681974705, 8.157183950347712692726002057248, 9.207834379220788354016903344416, 9.536451784505074313808134105535

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