↑

Properties

Label 1-1183-1183.1031-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.925 + 0.378i$
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  + (−0.799 − 0.600i)2-s + (−0.748 + 0.663i)3-s + (0.278 + 0.960i)4-s + (0.845 + 0.534i)5-s + (0.996 − 0.0804i)6-s + (0.354 − 0.935i)8-s + (0.120 − 0.992i)9-s + (−0.354 − 0.935i)10-s + (−0.120 − 0.992i)11-s + (−0.845 − 0.534i)12-s + (−0.987 + 0.160i)15-s + (−0.845 + 0.534i)16-s + (0.987 − 0.160i)17-s + (−0.692 + 0.721i)18-s − 19-s + (−0.278 + 0.960i)20-s + ⋯
L(s)  = 1  + (−0.799 − 0.600i)2-s + (−0.748 + 0.663i)3-s + (0.278 + 0.960i)4-s + (0.845 + 0.534i)5-s + (0.996 − 0.0804i)6-s + (0.354 − 0.935i)8-s + (0.120 − 0.992i)9-s + (−0.354 − 0.935i)10-s + (−0.120 − 0.992i)11-s + (−0.845 − 0.534i)12-s + (−0.987 + 0.160i)15-s + (−0.845 + 0.534i)16-s + (0.987 − 0.160i)17-s + (−0.692 + 0.721i)18-s − 19-s + (−0.278 + 0.960i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.925 + 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8465367933 + 0.1666224229i\)
\(L(\frac12)\) \(\approx\) \(0.8465367933 + 0.1666224229i\)
\(L(1)\) \(\approx\) \(0.6890511689 + 0.02744035810i\)
\(L(1)\) \(\approx\) \(0.6890511689 + 0.02744035810i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.799 - 0.600i)T \)
3 \( 1 + (-0.748 + 0.663i)T \)
5 \( 1 + (0.845 + 0.534i)T \)
11 \( 1 + (-0.120 - 0.992i)T \)
17 \( 1 + (0.987 - 0.160i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.919 + 0.391i)T \)
31 \( 1 + (0.996 - 0.0804i)T \)
37 \( 1 + (0.996 - 0.0804i)T \)
41 \( 1 + (-0.948 - 0.316i)T \)
43 \( 1 + (0.428 + 0.903i)T \)
47 \( 1 + (-0.692 - 0.721i)T \)
53 \( 1 + (-0.632 - 0.774i)T \)
59 \( 1 + (0.845 + 0.534i)T \)
61 \( 1 + (-0.354 - 0.935i)T \)
67 \( 1 + (0.970 - 0.239i)T \)
71 \( 1 + (0.200 + 0.979i)T \)
73 \( 1 + (0.919 + 0.391i)T \)
79 \( 1 + (0.692 + 0.721i)T \)
83 \( 1 + (0.748 + 0.663i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.845 - 0.534i)T \)
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

−20.98623572902223806734113115094, −20.343952044742383073873864614387, −19.345668009699884939944840500118, −18.62040870108107012172320470615, −17.98739168739513935947718272302, −17.23038016038089400514865138415, −16.87319675719165909649452878900, −16.120303589518883047969982934697, −15.05541340646784201917973126669, −14.26416358255383103106639711251, −13.33989015652529298076547416320, −12.563788861292752699959461651786, −11.81932151727116612663727376846, −10.63320031262501050214811251959, −10.119988951822683463670776109834, −9.31912094030749129125913264195, −8.23385671244194935627966880979, −7.64986854601583909713728672842, −6.53853722118660556035825699303, −6.126993797811029678978643835031, −5.18614093258846328285032238425, −4.50181921802538329658725271664, −2.34640490314135597751394963510, −1.74985246086428226691875215256, −0.70405609761798157723029182913, 0.83692231249873074520060149126, 1.99265931528456371802288022105, 3.14865680497147505917344987013, 3.777074574504955365307166056492, 5.10845422254457189085663168459, 6.02304105801921065223230325993, 6.68991754087780663215579576749, 7.855800799486903704380971706099, 8.8072270687040930052505246914, 9.78590279875418047601721148005, 10.06359539042820319177449999508, 11.103757726878027241379344747639, 11.3850839801222101611542374188, 12.49589038391791257199926380821, 13.29944589356265864987128000924, 14.29491170510012444359821731194, 15.2623965640329980899449000666, 16.20117253300195429661666417012, 16.86083993089894361083443951475, 17.36754720626912974863228885720, 18.26450567210496924480720992587, 18.73952030171267085044265765145, 19.645599312965214065326157954123, 20.77342313335373593706753064677, 21.315600496715999971539209571050

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