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Properties

Label 1-1183-1183.849-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.747 + 0.664i$
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  + (−0.428 − 0.903i)2-s + (0.885 − 0.464i)3-s + (−0.632 + 0.774i)4-s + (0.200 − 0.979i)5-s + (−0.799 − 0.600i)6-s + (0.970 + 0.239i)8-s + (0.568 − 0.822i)9-s + (−0.970 + 0.239i)10-s + (−0.568 − 0.822i)11-s + (−0.200 + 0.979i)12-s + (−0.278 − 0.960i)15-s + (−0.200 − 0.979i)16-s + (0.278 + 0.960i)17-s + (−0.987 − 0.160i)18-s − 19-s + (0.632 + 0.774i)20-s + ⋯
L(s)  = 1  + (−0.428 − 0.903i)2-s + (0.885 − 0.464i)3-s + (−0.632 + 0.774i)4-s + (0.200 − 0.979i)5-s + (−0.799 − 0.600i)6-s + (0.970 + 0.239i)8-s + (0.568 − 0.822i)9-s + (−0.970 + 0.239i)10-s + (−0.568 − 0.822i)11-s + (−0.200 + 0.979i)12-s + (−0.278 − 0.960i)15-s + (−0.200 − 0.979i)16-s + (0.278 + 0.960i)17-s + (−0.987 − 0.160i)18-s − 19-s + (0.632 + 0.774i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.747 + 0.664i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3286758675 - 0.8644466790i\)
\(L(\frac12)\) \(\approx\) \(-0.3286758675 - 0.8644466790i\)
\(L(1)\) \(\approx\) \(0.6064935099 - 0.7011629301i\)
\(L(1)\) \(\approx\) \(0.6064935099 - 0.7011629301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.428 - 0.903i)T \)
3 \( 1 + (0.885 - 0.464i)T \)
5 \( 1 + (0.200 - 0.979i)T \)
11 \( 1 + (-0.568 - 0.822i)T \)
17 \( 1 + (0.278 + 0.960i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.996 - 0.0804i)T \)
31 \( 1 + (-0.799 - 0.600i)T \)
37 \( 1 + (-0.799 - 0.600i)T \)
41 \( 1 + (0.845 + 0.534i)T \)
43 \( 1 + (-0.919 - 0.391i)T \)
47 \( 1 + (-0.987 + 0.160i)T \)
53 \( 1 + (0.692 - 0.721i)T \)
59 \( 1 + (0.200 - 0.979i)T \)
61 \( 1 + (-0.970 + 0.239i)T \)
67 \( 1 + (0.354 - 0.935i)T \)
71 \( 1 + (0.0402 - 0.999i)T \)
73 \( 1 + (0.996 - 0.0804i)T \)
79 \( 1 + (0.987 - 0.160i)T \)
83 \( 1 + (-0.885 - 0.464i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.200 + 0.979i)T \)
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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

−21.76388324656278582838296396271, −20.86460398493195021021534082807, −20.03871455102778924978693677534, −19.2436328512198529623062924939, −18.42387801622196679916039541205, −18.100098197907887284176098347079, −16.95019154049872200125851236326, −16.17615177044739692447065622908, −15.34101025709199766485813508082, −14.85559287251438849017282932013, −14.23309348066773092940988309244, −13.51749632128355905949009636569, −12.61420089415479240075972971927, −11.06743906924026019889080356610, −10.320821738301737973160145535877, −9.81859455764612543019471619753, −8.96701588690372873071051517019, −8.09265050353227023528695645856, −7.303667347994350531581219789661, −6.76550929545102696574077806997, −5.57018484709279667393545006932, −4.69273200744720965377279566423, −3.76678031330665601604881153620, −2.60277745711679907144625039782, −1.79620958001101788633364361818, 0.35446778865084565848882035130, 1.65045670045112948389649968752, 2.101840528821188756475114694183, 3.43087471662577361038455178952, 3.94365032179287526128419926833, 5.15709766613985437876028006788, 6.21075186962435181964320381385, 7.6782930262463238821803589532, 8.125615368370326112604008831953, 8.87222297889583915984114133500, 9.494725092644481837302377525191, 10.38842277166353007785284945119, 11.340720267486044619943166258064, 12.320492380860348937668814112958, 12.993808187342179376574800075830, 13.35377617430532066061758001425, 14.261217153710660465786112012715, 15.28377043831728179478078774528, 16.35041976581988194262955987251, 17.02989100400038860421896463472, 17.88791490904860711817708381041, 18.637922031182393104359567671046, 19.431097049609547253680373862, 19.82029565850597976217168065214, 20.7383310970981361241208070874

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