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Properties

Label 1-1183-1183.1032-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.927 + 0.374i$
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  + (0.903 − 0.428i)2-s + (0.845 + 0.534i)3-s + (0.632 − 0.774i)4-s + (0.316 + 0.948i)5-s + (0.992 + 0.120i)6-s + (0.239 − 0.970i)8-s + (0.428 + 0.903i)9-s + (0.692 + 0.721i)10-s + (0.903 + 0.428i)11-s + (0.948 − 0.316i)12-s + (−0.239 + 0.970i)15-s + (−0.200 − 0.979i)16-s + (0.278 + 0.960i)17-s + (0.774 + 0.632i)18-s + (−0.866 − 0.5i)19-s + (0.935 + 0.354i)20-s + ⋯
L(s)  = 1  + (0.903 − 0.428i)2-s + (0.845 + 0.534i)3-s + (0.632 − 0.774i)4-s + (0.316 + 0.948i)5-s + (0.992 + 0.120i)6-s + (0.239 − 0.970i)8-s + (0.428 + 0.903i)9-s + (0.692 + 0.721i)10-s + (0.903 + 0.428i)11-s + (0.948 − 0.316i)12-s + (−0.239 + 0.970i)15-s + (−0.200 − 0.979i)16-s + (0.278 + 0.960i)17-s + (0.774 + 0.632i)18-s + (−0.866 − 0.5i)19-s + (0.935 + 0.354i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1032, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.927 + 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.031543244 + 0.7835062466i\)
\(L(\frac12)\) \(\approx\) \(4.031543244 + 0.7835062466i\)
\(L(1)\) \(\approx\) \(2.530838288 + 0.1718581696i\)
\(L(1)\) \(\approx\) \(2.530838288 + 0.1718581696i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.903 - 0.428i)T \)
3 \( 1 + (0.845 + 0.534i)T \)
5 \( 1 + (0.316 + 0.948i)T \)
11 \( 1 + (0.903 + 0.428i)T \)
17 \( 1 + (0.278 + 0.960i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.568 - 0.822i)T \)
31 \( 1 + (-0.391 - 0.919i)T \)
37 \( 1 + (-0.600 + 0.799i)T \)
41 \( 1 + (0.464 + 0.885i)T \)
43 \( 1 + (-0.120 - 0.992i)T \)
47 \( 1 + (-0.774 + 0.632i)T \)
53 \( 1 + (0.278 + 0.960i)T \)
59 \( 1 + (0.979 + 0.200i)T \)
61 \( 1 + (-0.278 + 0.960i)T \)
67 \( 1 + (-0.160 - 0.987i)T \)
71 \( 1 + (-0.464 - 0.885i)T \)
73 \( 1 + (-0.903 - 0.428i)T \)
79 \( 1 + (-0.632 - 0.774i)T \)
83 \( 1 + (-0.464 + 0.885i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.663 - 0.748i)T \)
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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

−21.26179307740858866341225473603, −20.50172811265072819341887465128, −19.79784624271729015120452440012, −19.16718454575084520188349280079, −17.8799145384758447632309370129, −17.27168035607047696882139573513, −16.29435787671451024982913643621, −15.82173908209160060079380704287, −14.53373424586650959594002566464, −14.304738312264489059471764813041, −13.40061515698447798599227689304, −12.786841699451287121900929168593, −12.133669187886576600668024310134, −11.35095837205536362320993450148, −9.92821747886649345486935941304, −8.83550356646974939745004080026, −8.54595077288832712044034417811, −7.41181813788898778947326748704, −6.73993761839208607005309591324, −5.7920574840773250264621900445, −4.93273777929253133153233569452, −3.91675636141209454205002636800, −3.21703364440158243953304054701, −2.06279775335216291406460117444, −1.210955629898527262737348641154, 1.58606787338484576380025074146, 2.385271020313760492256125322833, 3.14747160456141079177361475209, 4.050780766676978685739867232449, 4.62302384914126972247946305736, 5.95452349838189539342583438625, 6.625932161231776790128636043519, 7.51681021302726851231232652447, 8.70462831795901990405338575694, 9.66268732079606725465145189818, 10.33428939603127845877062395469, 10.93022473677330334314355541117, 11.87609777814271307286426680816, 12.90457177433122765500344102966, 13.59270080122364522066493497996, 14.374452272553657175909945662666, 15.03599717078984529396345923014, 15.204662183552471594989549527678, 16.49229795008350917124398871456, 17.32926459309097230581928896862, 18.59214608542313554160047882319, 19.289734944659796597561203898725, 19.70602674886659631800937515128, 20.747279545366629189104923139174, 21.24791664651146163637232469001

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