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Properties

Label 1-1183-1183.110-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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.774 + 0.632i)2-s + (−0.568 − 0.822i)3-s + (0.200 − 0.979i)4-s + (−0.391 − 0.919i)5-s + (0.960 + 0.278i)6-s + (0.464 + 0.885i)8-s + (−0.354 + 0.935i)9-s + (0.885 + 0.464i)10-s + (0.935 − 0.354i)11-s + (−0.919 + 0.391i)12-s + (−0.534 + 0.845i)15-s + (−0.919 − 0.391i)16-s + (−0.845 − 0.534i)17-s + (−0.316 − 0.948i)18-s + i·19-s + (−0.979 + 0.200i)20-s + ⋯
L(s)  = 1  + (−0.774 + 0.632i)2-s + (−0.568 − 0.822i)3-s + (0.200 − 0.979i)4-s + (−0.391 − 0.919i)5-s + (0.960 + 0.278i)6-s + (0.464 + 0.885i)8-s + (−0.354 + 0.935i)9-s + (0.885 + 0.464i)10-s + (0.935 − 0.354i)11-s + (−0.919 + 0.391i)12-s + (−0.534 + 0.845i)15-s + (−0.919 − 0.391i)16-s + (−0.845 − 0.534i)17-s + (−0.316 − 0.948i)18-s + i·19-s + (−0.979 + 0.200i)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: $\chi_{1183} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7140450929 - 0.3028036323i\)
\(L(\frac12)\) \(\approx\) \(0.7140450929 - 0.3028036323i\)
\(L(1)\) \(\approx\) \(0.6228564618 - 0.1154041255i\)
\(L(1)\) \(\approx\) \(0.6228564618 - 0.1154041255i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.774 + 0.632i)T \)
3 \( 1 + (-0.568 - 0.822i)T \)
5 \( 1 + (-0.391 - 0.919i)T \)
11 \( 1 + (0.935 - 0.354i)T \)
17 \( 1 + (-0.845 - 0.534i)T \)
19 \( 1 + iT \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.987 - 0.160i)T \)
31 \( 1 + (0.960 + 0.278i)T \)
37 \( 1 + (0.960 + 0.278i)T \)
41 \( 1 + (0.903 + 0.428i)T \)
43 \( 1 + (-0.692 + 0.721i)T \)
47 \( 1 + (0.316 - 0.948i)T \)
53 \( 1 + (-0.0402 + 0.999i)T \)
59 \( 1 + (0.391 + 0.919i)T \)
61 \( 1 + (-0.885 - 0.464i)T \)
67 \( 1 + (0.663 + 0.748i)T \)
71 \( 1 + (0.0804 + 0.996i)T \)
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 \( 1 + (0.391 - 0.919i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.52316051585439931524524421805, −20.44568491792657489582931473491, −19.645980655770475038834712472350, −19.24217135849740319553768668349, −18.02323244167144116725246748034, −17.570525770187036039737935955130, −16.983069349093280433325786699312, −15.85413779854778180484517404444, −15.42393685994891040460170584022, −14.54434663594009543219980355662, −13.427748322952795014788102681252, −12.29298634084804383063989941284, −11.58221478027972865230783182193, −11.059792301058096278513610411352, −10.408722004057594408006062228839, −9.521647385443429373239573399, −8.95263842726452409495067432211, −7.85138650883326413579526896382, −6.783549853762992712886014688895, −6.34428963837460692329746833985, −4.73767295170528579133255851448, −3.994199824896999549667693435017, −3.2139012341095864863209604460, −2.23610637299743863778156294735, −0.78278700446403532520235536576, 0.733340430042777740221727327532, 1.33747321838500120042357916265, 2.550452646654396544302138596502, 4.325865163000428971155560557549, 5.03723377455086799798679529400, 6.11814587009537323111758088014, 6.59350482171206304788157236817, 7.602245896004053476972956024071, 8.34472741948047462972408466362, 8.91513252125295566455968106773, 9.931484892454856103703567311302, 10.98629891865254121154770538698, 11.70196874593876108611672564509, 12.36193389651008714708242808567, 13.41620310093806778200078495601, 14.10084010318103373260369351661, 15.09415929050709881031969909786, 16.12586692825161004647744560303, 16.60231852315962991021542671953, 17.18111462992892174593470290803, 17.98619848779785832959135015473, 18.69817572535972323733090398710, 19.54125668848524912933976686542, 19.92050419822041651878034883546, 20.891557606927147579607684332323

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