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Properties

Label 1-1183-1183.459-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.201 + 0.979i$
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  + (−0.568 − 0.822i)2-s + (−0.845 + 0.534i)3-s + (−0.354 + 0.935i)4-s + (0.200 + 0.979i)5-s + (0.919 + 0.391i)6-s + (0.970 − 0.239i)8-s + (0.428 − 0.903i)9-s + (0.692 − 0.721i)10-s + (−0.428 − 0.903i)11-s + (−0.200 − 0.979i)12-s + (−0.692 − 0.721i)15-s + (−0.748 − 0.663i)16-s + (−0.970 + 0.239i)17-s + (−0.987 + 0.160i)18-s + (0.5 + 0.866i)19-s + (−0.987 − 0.160i)20-s + ⋯
L(s)  = 1  + (−0.568 − 0.822i)2-s + (−0.845 + 0.534i)3-s + (−0.354 + 0.935i)4-s + (0.200 + 0.979i)5-s + (0.919 + 0.391i)6-s + (0.970 − 0.239i)8-s + (0.428 − 0.903i)9-s + (0.692 − 0.721i)10-s + (−0.428 − 0.903i)11-s + (−0.200 − 0.979i)12-s + (−0.692 − 0.721i)15-s + (−0.748 − 0.663i)16-s + (−0.970 + 0.239i)17-s + (−0.987 + 0.160i)18-s + (0.5 + 0.866i)19-s + (−0.987 − 0.160i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.201 + 0.979i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.201 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3155788246 + 0.3871802240i\)
\(L(\frac12)\) \(\approx\) \(0.3155788246 + 0.3871802240i\)
\(L(1)\) \(\approx\) \(0.5619693209 + 0.04456615706i\)
\(L(1)\) \(\approx\) \(0.5619693209 + 0.04456615706i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.568 - 0.822i)T \)
3 \( 1 + (-0.845 + 0.534i)T \)
5 \( 1 + (0.200 + 0.979i)T \)
11 \( 1 + (-0.428 - 0.903i)T \)
17 \( 1 + (-0.970 + 0.239i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.428 - 0.903i)T \)
31 \( 1 + (-0.799 + 0.600i)T \)
37 \( 1 + (-0.120 - 0.992i)T \)
41 \( 1 + (0.0402 + 0.999i)T \)
43 \( 1 + (0.799 + 0.600i)T \)
47 \( 1 + (-0.987 - 0.160i)T \)
53 \( 1 + (0.692 + 0.721i)T \)
59 \( 1 + (0.748 - 0.663i)T \)
61 \( 1 + (0.278 + 0.960i)T \)
67 \( 1 + (-0.987 - 0.160i)T \)
71 \( 1 + (0.845 - 0.534i)T \)
73 \( 1 + (0.996 + 0.0804i)T \)
79 \( 1 + (0.987 + 0.160i)T \)
83 \( 1 + (-0.885 + 0.464i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.948 + 0.316i)T \)
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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

−20.861888722097631297978657634227, −20.03350491380027695200441522764, −19.364772016496826187028047415950, −18.31842221106996958545793608335, −17.78723923951067920985775088717, −17.23809294963444553144494905943, −16.5122295916921377125640733621, −15.79036520380125562144574870821, −15.159778877145612886882620408886, −13.87222128568625337922358007902, −13.17732869375680894810555863159, −12.60031129351861734644561695018, −11.49861491066931521038705396447, −10.71696262311957064453336844287, −9.76602663919242214968912614587, −9.01771954038466622137576834920, −8.18347600489508509560673177628, −7.154016711488490033593703765788, −6.78451558994498091388257014100, −5.54281248323959482741364457095, −5.0565626810748583930151847834, −4.382079292033286460668053570494, −2.28785932312140798495322473293, −1.37399662274541149659491462412, −0.332376181220979335639482516215, 1.081191606193358795989836773, 2.38893383250408352018267065153, 3.307342530613458959587365964637, 4.032959100162643727690144085626, 5.18032450868430288479710035930, 6.143122754271910340197566855026, 7.00036720160416993837179473390, 7.97834879158818993775076073596, 9.043283114420025068638770244331, 9.80403353106890044041707911272, 10.625205577537604945625228757545, 11.04133246223386912341833592587, 11.6591448173724708987902546246, 12.67343514462100253188579562414, 13.47302217603605554329358522822, 14.429817759790216562331363490651, 15.44375938243399040981428462463, 16.25698259881229140723920649101, 16.9218715465534182542453672524, 17.949910894556609496186050947607, 18.13196928285677662147307814921, 19.083623523589954155894796095791, 19.76151566910585204944878821862, 21.065908441981318853344672713603, 21.2582762118302248598416782009

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