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Properties

Label 1-1183-1183.593-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.277 - 0.960i$
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  + (−0.960 + 0.278i)2-s + (−0.799 − 0.600i)3-s + (0.845 − 0.534i)4-s + (−0.0804 + 0.996i)5-s + (0.935 + 0.354i)6-s + (−0.663 + 0.748i)8-s + (0.278 + 0.960i)9-s + (−0.200 − 0.979i)10-s + (−0.960 − 0.278i)11-s + (−0.996 − 0.0804i)12-s + (0.663 − 0.748i)15-s + (0.428 − 0.903i)16-s + (0.948 − 0.316i)17-s + (−0.534 − 0.845i)18-s + (−0.866 + 0.5i)19-s + (0.464 + 0.885i)20-s + ⋯
L(s)  = 1  + (−0.960 + 0.278i)2-s + (−0.799 − 0.600i)3-s + (0.845 − 0.534i)4-s + (−0.0804 + 0.996i)5-s + (0.935 + 0.354i)6-s + (−0.663 + 0.748i)8-s + (0.278 + 0.960i)9-s + (−0.200 − 0.979i)10-s + (−0.960 − 0.278i)11-s + (−0.996 − 0.0804i)12-s + (0.663 − 0.748i)15-s + (0.428 − 0.903i)16-s + (0.948 − 0.316i)17-s + (−0.534 − 0.845i)18-s + (−0.866 + 0.5i)19-s + (0.464 + 0.885i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.277 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1163203352 - 0.1547496984i\)
\(L(\frac12)\) \(\approx\) \(0.1163203352 - 0.1547496984i\)
\(L(1)\) \(\approx\) \(0.4368097904 + 0.03277757804i\)
\(L(1)\) \(\approx\) \(0.4368097904 + 0.03277757804i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.960 + 0.278i)T \)
3 \( 1 + (-0.799 - 0.600i)T \)
5 \( 1 + (-0.0804 + 0.996i)T \)
11 \( 1 + (-0.960 - 0.278i)T \)
17 \( 1 + (0.948 - 0.316i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.970 - 0.239i)T \)
31 \( 1 + (-0.774 + 0.632i)T \)
37 \( 1 + (-0.160 - 0.987i)T \)
41 \( 1 + (-0.992 - 0.120i)T \)
43 \( 1 + (0.354 + 0.935i)T \)
47 \( 1 + (0.534 - 0.845i)T \)
53 \( 1 + (0.948 - 0.316i)T \)
59 \( 1 + (-0.903 + 0.428i)T \)
61 \( 1 + (-0.948 - 0.316i)T \)
67 \( 1 + (-0.999 - 0.0402i)T \)
71 \( 1 + (0.992 + 0.120i)T \)
73 \( 1 + (0.960 + 0.278i)T \)
79 \( 1 + (-0.845 - 0.534i)T \)
83 \( 1 + (0.992 - 0.120i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.822 - 0.568i)T \)
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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

−21.227417055457888655720043532914, −20.629387701644927030685358130245, −20.156716519854967777967619512853, −18.91713197829803261953760514960, −18.41159762355719070819674909729, −17.38770913310488096460291182170, −16.82693671027109864159905170772, −16.43949187509508634190648307193, −15.428406715238967768045415000957, −15.021091813981299950391776551747, −13.27582645621961705028421111992, −12.51685909262273552304124409474, −12.05152869944540486376193724933, −10.957569437232898078264657433541, −10.48255927501724220090560408397, −9.59346225937797712816795619533, −8.923324083616797996489721101792, −8.070917205992786288709770208633, −7.180222189976282486274184923087, −6.08706088114231745847474432751, −5.26301397224795613760819579033, −4.36567933534048075442447305135, −3.356404839449183134213471924001, −2.07177479031893748052459647833, −0.93275485972193435032064487956, 0.1469794540090728131799209097, 1.5702607170400919720544681060, 2.429320624562148434575741438302, 3.49531380167475711225515169934, 5.24132486593238790685070010596, 5.8104994336963592565573737603, 6.65917645137002612878843374313, 7.54633634338423253239869235433, 7.80012335973653712295900619972, 9.07835148965167788710706733752, 10.240275682651136775974540538515, 10.6260128043895794524752394831, 11.35948732828192139440491671409, 12.12234278920025044959876407610, 13.13401955777103733728313037502, 14.12405960454956248424741732577, 14.98628520333332524001103257816, 15.76107518525185421297931330090, 16.59061017438944940192891306660, 17.19887159623986820917016656067, 18.17702250093456561189450318122, 18.482299451625783120046513463587, 19.10868774016055069589974642607, 19.832376531010067482305748623036, 21.11979372075503479477713052554

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