↑

Properties

Label 2-1183-7.2-c1-0-49
Degree $2$
Conductor $1183$
Sign $0.722 + 0.691i$
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  + (−0.294 + 0.509i)2-s + (−0.386 − 0.670i)3-s + (0.826 + 1.43i)4-s + (−1.34 + 2.33i)5-s + 0.455·6-s + (−2.52 − 0.791i)7-s − 2.15·8-s + (1.20 − 2.07i)9-s + (−0.793 − 1.37i)10-s + (−1.34 − 2.33i)11-s + (0.639 − 1.10i)12-s + (1.14 − 1.05i)14-s + 2.08·15-s + (−1.02 + 1.76i)16-s + (0.237 + 0.411i)17-s + (0.706 + 1.22i)18-s + ⋯
L(s)  = 1  + (−0.208 + 0.360i)2-s + (−0.223 − 0.386i)3-s + (0.413 + 0.716i)4-s + (−0.602 + 1.04i)5-s + 0.185·6-s + (−0.954 − 0.298i)7-s − 0.760·8-s + (0.400 − 0.693i)9-s + (−0.250 − 0.434i)10-s + (−0.405 − 0.703i)11-s + (0.184 − 0.319i)12-s + (0.306 − 0.281i)14-s + 0.538·15-s + (−0.255 + 0.442i)16-s + (0.0576 + 0.0998i)17-s + (0.166 + 0.288i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7751333876\)
\(L(\frac12)\) \(\approx\) \(0.7751333876\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.52 + 0.791i)T \)
13 \( 1 \)
good2 \( 1 + (0.294 - 0.509i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.386 + 0.670i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.34 - 2.33i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.34 + 2.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.237 - 0.411i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.980 + 1.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.86 + 4.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.40T + 29T^{2} \)
31 \( 1 + (1.65 + 2.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.44 + 9.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + 9.43T + 43T^{2} \)
47 \( 1 + (4.49 - 7.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.15 - 3.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.96 - 8.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.98 + 8.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.70 + 2.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.27T + 71T^{2} \)
73 \( 1 + (2.10 + 3.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.39 + 12.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.99T + 83T^{2} \)
89 \( 1 + (-1.83 + 3.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.3T + 97T^{2} \)
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

−9.559727753838475134363293150917, −8.768543611374676258502416261120, −7.62227318202436805885574000349, −7.27477103778958352496260049564, −6.45966601723042752354888855499, −5.99931392720131165594495813997, −4.15907927315157499651691525434, −3.30286388458038292606712407965, −2.67086051832038612832242103407, −0.40668861861901552825852049643, 1.16965059117278590909206332865, 2.46431149484641679691479116281, 3.72389472430225818852058656170, 4.93067267104779018749109619263, 5.35199048013358765736287587860, 6.50537273319595719529443935226, 7.41570976533724360894599021776, 8.370508105196728500988416617264, 9.343301569547609424312219640620, 9.898787037732914304948294384964

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