↑

Properties

Label 1-1183-1183.502-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.695 + 0.718i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.160 + 0.987i)2-s + (0.996 − 0.0804i)3-s + (−0.948 + 0.316i)4-s + (0.391 + 0.919i)5-s + (0.239 + 0.970i)6-s + (−0.464 − 0.885i)8-s + (0.987 − 0.160i)9-s + (−0.845 + 0.534i)10-s + (0.160 − 0.987i)11-s + (−0.919 + 0.391i)12-s + (0.464 + 0.885i)15-s + (0.799 − 0.600i)16-s + (−0.0402 + 0.999i)17-s + (0.316 + 0.948i)18-s + (−0.866 + 0.5i)19-s + (−0.663 − 0.748i)20-s + ⋯
L(s)  = 1  + (0.160 + 0.987i)2-s + (0.996 − 0.0804i)3-s + (−0.948 + 0.316i)4-s + (0.391 + 0.919i)5-s + (0.239 + 0.970i)6-s + (−0.464 − 0.885i)8-s + (0.987 − 0.160i)9-s + (−0.845 + 0.534i)10-s + (0.160 − 0.987i)11-s + (−0.919 + 0.391i)12-s + (0.464 + 0.885i)15-s + (0.799 − 0.600i)16-s + (−0.0402 + 0.999i)17-s + (0.316 + 0.948i)18-s + (−0.866 + 0.5i)19-s + (−0.663 − 0.748i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.695 + 0.718i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (502, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.695 + 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8700675124 + 2.051717256i\)
\(L(\frac12)\) \(\approx\) \(0.8700675124 + 2.051717256i\)
\(L(1)\) \(\approx\) \(1.173903584 + 0.9874473817i\)
\(L(1)\) \(\approx\) \(1.173903584 + 0.9874473817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.160 + 0.987i)T \)
3 \( 1 + (0.996 - 0.0804i)T \)
5 \( 1 + (0.391 + 0.919i)T \)
11 \( 1 + (0.160 - 0.987i)T \)
17 \( 1 + (-0.0402 + 0.999i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.354 + 0.935i)T \)
31 \( 1 + (-0.960 - 0.278i)T \)
37 \( 1 + (0.721 - 0.692i)T \)
41 \( 1 + (0.822 - 0.568i)T \)
43 \( 1 + (0.970 + 0.239i)T \)
47 \( 1 + (-0.316 + 0.948i)T \)
53 \( 1 + (-0.0402 + 0.999i)T \)
59 \( 1 + (-0.600 + 0.799i)T \)
61 \( 1 + (0.0402 + 0.999i)T \)
67 \( 1 + (0.979 - 0.200i)T \)
71 \( 1 + (-0.822 + 0.568i)T \)
73 \( 1 + (-0.160 + 0.987i)T \)
79 \( 1 + (0.948 + 0.316i)T \)
83 \( 1 + (-0.822 - 0.568i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.992 - 0.120i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.7548596358782667608016134974, −20.28720696695084128258906291237, −19.70181633380025036784925376995, −18.84377317509002586259323058200, −18.05941784225606274527881375652, −17.27222553782866921161844862996, −16.31830664185364724450846452378, −15.23250836700296628812224675748, −14.56884795812255410613349820061, −13.7623005556346224854231973286, −12.94964294840469170688025963646, −12.649999443096739817774388235211, −11.618651241422729626238242237023, −10.56081409155923269114814154303, −9.63725814547176060267722201298, −9.27845787308905980785388293732, −8.505775568057046459299995395569, −7.60622851793908549643329679002, −6.378780962160014556186623427380, −4.935888942709721287566100232685, −4.59710856360372181604211198964, −3.63488579716767410259348760946, −2.421125489676835391551273226682, −1.988100277136756322262956930594, −0.78048184937825184930490701803, 1.4241742499863306431946125352, 2.68541651670493956049897497765, 3.57285581785331979513078839509, 4.16983876283062559675545220563, 5.69136353839602028828219687371, 6.193096708566466732153949400756, 7.23985419547192514134926009849, 7.75930712891893303437443814768, 8.79046366108631083971360063441, 9.27020976734043124391621891791, 10.33477163702216877251654612164, 11.10366341196675960965473849956, 12.662170988033918127183921750014, 13.12842510660687334831384176236, 14.09811619030430442704782014762, 14.485613112411467581672008206292, 15.11613334201190658106859536895, 15.9291783900664646309361740542, 16.82493218115530979997264392371, 17.65361162655549293859652167836, 18.49966045123611990306679986078, 19.054121141195141055494148929960, 19.67775165876176177813579766267, 21.108002169729293009429966141150, 21.54870107659617218149728926668

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