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Properties

Label 1-1183-1183.201-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.973 - 0.226i$
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  + (−0.316 − 0.948i)2-s + (0.354 − 0.935i)3-s + (−0.799 + 0.600i)4-s + (−0.960 + 0.278i)5-s + (−0.999 − 0.0402i)6-s + (0.822 + 0.568i)8-s + (−0.748 − 0.663i)9-s + (0.568 + 0.822i)10-s + (−0.663 − 0.748i)11-s + (0.278 + 0.960i)12-s + (−0.0804 + 0.996i)15-s + (0.278 − 0.960i)16-s + (−0.996 − 0.0804i)17-s + (−0.391 + 0.919i)18-s + i·19-s + (0.600 − 0.799i)20-s + ⋯
L(s)  = 1  + (−0.316 − 0.948i)2-s + (0.354 − 0.935i)3-s + (−0.799 + 0.600i)4-s + (−0.960 + 0.278i)5-s + (−0.999 − 0.0402i)6-s + (0.822 + 0.568i)8-s + (−0.748 − 0.663i)9-s + (0.568 + 0.822i)10-s + (−0.663 − 0.748i)11-s + (0.278 + 0.960i)12-s + (−0.0804 + 0.996i)15-s + (0.278 − 0.960i)16-s + (−0.996 − 0.0804i)17-s + (−0.391 + 0.919i)18-s + i·19-s + (0.600 − 0.799i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.973 - 0.226i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.973 - 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4755680937 - 0.05468973561i\)
\(L(\frac12)\) \(\approx\) \(0.4755680937 - 0.05468973561i\)
\(L(1)\) \(\approx\) \(0.5209655967 - 0.3695627809i\)
\(L(1)\) \(\approx\) \(0.5209655967 - 0.3695627809i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.316 - 0.948i)T \)
3 \( 1 + (0.354 - 0.935i)T \)
5 \( 1 + (-0.960 + 0.278i)T \)
11 \( 1 + (-0.663 - 0.748i)T \)
17 \( 1 + (-0.996 - 0.0804i)T \)
19 \( 1 + iT \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.200 + 0.979i)T \)
31 \( 1 + (-0.999 - 0.0402i)T \)
37 \( 1 + (-0.999 - 0.0402i)T \)
41 \( 1 + (0.160 + 0.987i)T \)
43 \( 1 + (0.845 - 0.534i)T \)
47 \( 1 + (0.391 + 0.919i)T \)
53 \( 1 + (0.428 + 0.903i)T \)
59 \( 1 + (0.960 - 0.278i)T \)
61 \( 1 + (-0.568 - 0.822i)T \)
67 \( 1 + (-0.992 - 0.120i)T \)
71 \( 1 + (-0.774 + 0.632i)T \)
73 \( 1 + (-0.979 + 0.200i)T \)
79 \( 1 + (-0.919 + 0.391i)T \)
83 \( 1 + (0.935 - 0.354i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.960 + 0.278i)T \)
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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

−21.20082801997758227852899189138, −20.37316615244987783196486006232, −19.60172173027353934606912804098, −19.1348931625054619263101259142, −17.89329680072693929040174005154, −17.3180970588209593559958745628, −16.37328831324354020456500919222, −15.693311602300169268204005032728, −15.29907787528310060381353512838, −14.736160852311381415490923167764, −13.558324449856044766288464119183, −12.99188610502339399739250038241, −11.652971910237525989265296948232, −10.81268798225642305374561884997, −10.06186650290989676810042760844, −8.95338856763643171053115265908, −8.77567304207385800222547535380, −7.53416793316093182564670663659, −7.20507905080311630036603418949, −5.752835475318846707591970312759, −4.886949471071773847827624301314, −4.3610800958835231133473653570, −3.45178384935197410456685474660, −2.102190427179519919336787431884, −0.268036893478532309889564735970, 0.87863836811554503600063920384, 2.05268311678104412535485700402, 2.96030359110560434561673938836, 3.587765845753984215539831572426, 4.65179539709376716834060513226, 5.89836561154441934910200864987, 7.11230154480389637375736301220, 7.71873914130774735464993923476, 8.58887614694061194117320821261, 8.99368665369178180485754208694, 10.48075336853371154108197911701, 11.01072354533397532240601603973, 11.79691773188050142749955633624, 12.599998615672750030519846215874, 13.08248752648765955759914401569, 14.09075320892015740177724413491, 14.6932884980177717412143905572, 15.895584371842807972092830942036, 16.70508190453544355749958337398, 17.72661588371011052495536265224, 18.54392557193336873513589879933, 18.81268937161058885300313506974, 19.586762158767486427179418112677, 20.309978831249440705634934356654, 20.82094729546385535086240265685

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