↑

Properties

Label 2-1183-13.12-c1-0-44
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  + 0.470i·2-s + 2.24·3-s + 1.77·4-s + 0.529i·5-s + 1.05i·6-s + i·7-s + 1.77i·8-s + 2.05·9-s − 0.249·10-s + 2.24i·11-s + 4.00·12-s − 0.470·14-s + 1.19i·15-s + 2.71·16-s + 1.30·17-s + 0.968i·18-s + ⋯
L(s)  = 1  + 0.332i·2-s + 1.29·3-s + 0.889·4-s + 0.236i·5-s + 0.432i·6-s + 0.377i·7-s + 0.628i·8-s + 0.686·9-s − 0.0787·10-s + 0.678i·11-s + 1.15·12-s − 0.125·14-s + 0.307i·15-s + 0.679·16-s + 0.317·17-s + 0.228i·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

\(L(1)\) \(\approx\) \(3.132776763\)
\(L(\frac12)\) \(\approx\) \(3.132776763\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.470iT - 2T^{2} \)
3 \( 1 - 2.24T + 3T^{2} \)
5 \( 1 - 0.529iT - 5T^{2} \)
11 \( 1 - 2.24iT - 11T^{2} \)
17 \( 1 - 1.30T + 17T^{2} \)
19 \( 1 + 1.47iT - 19T^{2} \)
23 \( 1 + 5.83T + 23T^{2} \)
29 \( 1 - 5.22T + 29T^{2} \)
31 \( 1 + 7.02iT - 31T^{2} \)
37 \( 1 - 2.36iT - 37T^{2} \)
41 \( 1 - 6.49iT - 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + 8.58iT - 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 12.1iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 15.9iT - 67T^{2} \)
71 \( 1 - 1.19iT - 71T^{2} \)
73 \( 1 + 7.64iT - 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 + 16.3iT - 83T^{2} \)
89 \( 1 + 6.91iT - 89T^{2} \)
97 \( 1 + 3.47iT - 97T^{2} \)
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

−9.887782278105470001822897132035, −8.866036787396707621425486871118, −8.184013720465003958783355378792, −7.54430856265578919930218239166, −6.73177600191198475248351549826, −5.88996767872007662097333976877, −4.69361724017447109461350809707, −3.41974913888711034207875848598, −2.61487885678144888221684040194, −1.84189721348256383154906182178, 1.26392883376155372841798869972, 2.41668334275630003509019209442, 3.25501478348445739922291232627, 3.96779791187698894360331462799, 5.40044269367766515737729026998, 6.49321301825957342735992186696, 7.27856569877303025507833125105, 8.215850367911212569711345862033, 8.606819629701490195496291865280, 9.735916640838168686776179772869

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