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Properties

Label 1-1183-1183.538-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.994 - 0.102i$
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  + (0.663 + 0.748i)2-s + (0.354 + 0.935i)3-s + (−0.120 + 0.992i)4-s + (−0.239 − 0.970i)5-s + (−0.464 + 0.885i)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 + (−0.970 + 0.239i)12-s + (0.822 − 0.568i)15-s + (−0.970 − 0.239i)16-s + (0.568 + 0.822i)17-s + (−0.992 − 0.120i)18-s + i·19-s + (0.992 − 0.120i)20-s + ⋯
L(s)  = 1  + (0.663 + 0.748i)2-s + (0.354 + 0.935i)3-s + (−0.120 + 0.992i)4-s + (−0.239 − 0.970i)5-s + (−0.464 + 0.885i)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 + (−0.970 + 0.239i)12-s + (0.822 − 0.568i)15-s + (−0.970 − 0.239i)16-s + (0.568 + 0.822i)17-s + (−0.992 − 0.120i)18-s + i·19-s + (0.992 − 0.120i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.994 - 0.102i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.994 - 0.102i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08521513531 + 1.665491149i\)
\(L(\frac12)\) \(\approx\) \(-0.08521513531 + 1.665491149i\)
\(L(1)\) \(\approx\) \(0.9509777713 + 0.9868165541i\)
\(L(1)\) \(\approx\) \(0.9509777713 + 0.9868165541i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.663 + 0.748i)T \)
3 \( 1 + (0.354 + 0.935i)T \)
5 \( 1 + (-0.239 - 0.970i)T \)
11 \( 1 + (0.663 - 0.748i)T \)
17 \( 1 + (0.568 + 0.822i)T \)
19 \( 1 + iT \)
23 \( 1 - T \)
29 \( 1 + (-0.748 + 0.663i)T \)
31 \( 1 + (-0.464 + 0.885i)T \)
37 \( 1 + (-0.464 + 0.885i)T \)
41 \( 1 + (0.935 - 0.354i)T \)
43 \( 1 + (-0.885 + 0.464i)T \)
47 \( 1 + (0.992 - 0.120i)T \)
53 \( 1 + (0.568 + 0.822i)T \)
59 \( 1 + (0.239 + 0.970i)T \)
61 \( 1 + (-0.568 + 0.822i)T \)
67 \( 1 + (0.992 - 0.120i)T \)
71 \( 1 + (-0.935 + 0.354i)T \)
73 \( 1 + (-0.663 + 0.748i)T \)
79 \( 1 + (0.120 + 0.992i)T \)
83 \( 1 + (-0.935 - 0.354i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.239 - 0.970i)T \)
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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

−20.58944603327803175130554826698, −20.05388044081793109056571135450, −19.34756498689980746922053264940, −18.705302491526411825676031028917, −18.06517480377639488270600108056, −17.35269867103656027578163616266, −15.87768962764486674697525691291, −14.964320144982734636421864340079, −14.47961235072879866531412541739, −13.78206591873213396532509323624, −13.06675592618074907328316619605, −12.06971190014637385321715592629, −11.67180088831704527933911127825, −10.85982804974298602593707781911, −9.74708961652696228553736748751, −9.18089881461467392522378863936, −7.78840683315725515581996198858, −7.07340580392228063780188756848, −6.357919442069511056095344383280, −5.46902072398138816606553445091, −4.158350004627688077902345450597, −3.42683608967145525699718208596, −2.4377772324810773888111753684, −1.90318613467541470776743575235, −0.48499742275299468516992397739, 1.59262783071844320000266587400, 3.12577585320577860700325793183, 3.883116285063367203325316889217, 4.37137934046463450021765542927, 5.613197504789514284270720529776, 5.8024198476753542186076324222, 7.29796101914141054650949769280, 8.32988605707028033375292069175, 8.60450073423557722954183580121, 9.53271839042570711499349950750, 10.56832190784134396155410982410, 11.68914322214237068942061492271, 12.29278129943711449741068279498, 13.22264506329161654632405206471, 14.11519961580283184168133038463, 14.58877306941672650169853941041, 15.51360562925235691431679341140, 16.21663329757441985324964057571, 16.724561890565230405484569786080, 17.2105662023873579854017001977, 18.506233260751650586410399117507, 19.63731156747602325968434538534, 20.25618157109178093868641179438, 21.07823748569536482789390919993, 21.60349894839788388302435416740

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