↑

Properties

Label 2-1183-13.12-c1-0-48
Degree $2$
Conductor $1183$
Sign $0.554 - 0.832i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.74i·2-s + 1.36·3-s − 5.51·4-s − 0.741i·5-s + 3.74i·6-s − i·7-s − 9.63i·8-s − 1.13·9-s + 2.03·10-s − 1.36i·11-s − 7.52·12-s + 2.74·14-s − 1.01i·15-s + 15.3·16-s + 4.14·17-s − 3.11i·18-s + ⋯
L(s)  = 1  + 1.93i·2-s + 0.787·3-s − 2.75·4-s − 0.331i·5-s + 1.52i·6-s − 0.377i·7-s − 3.40i·8-s − 0.379·9-s + 0.642·10-s − 0.411i·11-s − 2.17·12-s + 0.732·14-s − 0.261i·15-s + 3.84·16-s + 1.00·17-s − 0.734i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.554 - 0.832i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.501877181\)
\(L(\frac12)\) \(\approx\) \(1.501877181\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + iT \)
13 \( 1 \)
good2 \( 1 - 2.74iT - 2T^{2} \)
3 \( 1 - 1.36T + 3T^{2} \)
5 \( 1 + 0.741iT - 5T^{2} \)
11 \( 1 + 1.36iT - 11T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 + 7.26iT - 19T^{2} \)
23 \( 1 - 2.33T + 23T^{2} \)
29 \( 1 + 0.407T + 29T^{2} \)
31 \( 1 - 2.77iT - 31T^{2} \)
37 \( 1 + 6.10iT - 37T^{2} \)
41 \( 1 + 1.25iT - 41T^{2} \)
43 \( 1 - 1.74T + 43T^{2} \)
47 \( 1 + 5.85iT - 47T^{2} \)
53 \( 1 - 4.56T + 53T^{2} \)
59 \( 1 + 10.9iT - 59T^{2} \)
61 \( 1 - 6.52T + 61T^{2} \)
67 \( 1 - 13.7iT - 67T^{2} \)
71 \( 1 - 4.81iT - 71T^{2} \)
73 \( 1 - 6.06iT - 73T^{2} \)
79 \( 1 + 9.12T + 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 + 1.76iT - 89T^{2} \)
97 \( 1 + 9.53iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.300105016149893515582764327251, −8.777689826951414056369573842215, −8.270761049188224985315654744052, −7.33581441693717967706113588070, −6.84054751460106176264304227433, −5.67224847027577355642295521228, −5.09080096387631121560173048332, −4.04588260064449802515949667203, −3.04262603085237798912218088604, −0.66656359102685495761752157929, 1.34879991423550882966259992644, 2.43550793323338630804984011593, 3.15641269659373116558184295480, 3.86856630570827302328373768268, 5.01410427953883751394008353176, 5.94874823631601856601696042929, 7.69173546300301187815490888207, 8.339346879486269221094272816785, 9.112445517268380482507755324952, 9.788015242048556345398511485802

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