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Properties

Label 1-1183-1183.604-r1-0-0
Degree $1$
Conductor $1183$
Sign $0.673 - 0.739i$
Analytic cond. $127.131$
Root an. cond. $127.131$
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  + (0.160 − 0.987i)2-s + (0.568 − 0.822i)3-s + (−0.948 − 0.316i)4-s + (−0.600 − 0.799i)5-s + (−0.721 − 0.692i)6-s + (−0.464 + 0.885i)8-s + (−0.354 − 0.935i)9-s + (−0.885 + 0.464i)10-s + (−0.935 − 0.354i)11-s + (−0.799 + 0.600i)12-s + (−0.999 + 0.0402i)15-s + (0.799 + 0.600i)16-s + (0.0402 + 0.999i)17-s + (−0.979 + 0.200i)18-s − i·19-s + (0.316 + 0.948i)20-s + ⋯
L(s)  = 1  + (0.160 − 0.987i)2-s + (0.568 − 0.822i)3-s + (−0.948 − 0.316i)4-s + (−0.600 − 0.799i)5-s + (−0.721 − 0.692i)6-s + (−0.464 + 0.885i)8-s + (−0.354 − 0.935i)9-s + (−0.885 + 0.464i)10-s + (−0.935 − 0.354i)11-s + (−0.799 + 0.600i)12-s + (−0.999 + 0.0402i)15-s + (0.799 + 0.600i)16-s + (0.0402 + 0.999i)17-s + (−0.979 + 0.200i)18-s − i·19-s + (0.316 + 0.948i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.673 - 0.739i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ 0.673 - 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6045721720 - 0.2670934299i\)
\(L(\frac12)\) \(\approx\) \(0.6045721720 - 0.2670934299i\)
\(L(1)\) \(\approx\) \(0.5356204886 - 0.7119940800i\)
\(L(1)\) \(\approx\) \(0.5356204886 - 0.7119940800i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.160 - 0.987i)T \)
3 \( 1 + (0.568 - 0.822i)T \)
5 \( 1 + (-0.600 - 0.799i)T \)
11 \( 1 + (-0.935 - 0.354i)T \)
17 \( 1 + (0.0402 + 0.999i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.632 + 0.774i)T \)
31 \( 1 + (-0.721 - 0.692i)T \)
37 \( 1 + (0.721 + 0.692i)T \)
41 \( 1 + (-0.0804 + 0.996i)T \)
43 \( 1 + (-0.278 + 0.960i)T \)
47 \( 1 + (-0.979 - 0.200i)T \)
53 \( 1 + (-0.845 + 0.534i)T \)
59 \( 1 + (0.600 + 0.799i)T \)
61 \( 1 + (0.885 - 0.464i)T \)
67 \( 1 + (-0.663 + 0.748i)T \)
71 \( 1 + (0.903 - 0.428i)T \)
73 \( 1 + (0.774 - 0.632i)T \)
79 \( 1 + (-0.200 + 0.979i)T \)
83 \( 1 + (0.822 - 0.568i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.600 - 0.799i)T \)
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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

−21.19046534101844251896171126841, −20.55338014794387632960449393494, −19.46045048425688447301341866507, −18.72785985287744241797097534283, −18.08239507124921300082316336933, −17.08751137058706822563894670952, −16.05653562892034029222071085491, −15.797055405120885663040386856504, −14.962267664374603593025572892286, −14.42798981847069506831182232308, −13.68432035986418682107852235386, −12.80869826979907904894763376265, −11.66846231179876529048280318614, −10.71305593395756584839132722999, −9.88979548672327557440299196353, −9.21141058610980774112632846204, −8.08859816700431536070495828186, −7.65042125105108651452995031077, −6.86598867912741616672399456439, −5.57606319101566881648057446899, −4.97846174096391075204430345002, −3.8728388972194692538549024398, −3.35549026295555222324843996168, −2.277906520376968655822851777739, −0.15673638184016103256792945474, 0.74849768382883226468812259155, 1.63755777580656704844530946813, 2.683427246428645683089607289065, 3.43314703475587011525757882895, 4.44598819637290596553917672126, 5.296920392334466544273083959032, 6.3670035216039820207292997125, 7.66254542298409372894451239458, 8.310762051754844719059822025052, 8.91564668557453412209363880492, 9.78425370570716729659154674362, 11.00294039064350039447582817128, 11.50764404685362033930158343526, 12.655336609582815463398402975741, 12.92105801114179839458614680963, 13.47766982819475665817464287229, 14.68851789776755427195965513506, 15.15140427490792835074924812264, 16.4143266796471145619953530691, 17.27878053015493764156756438964, 18.25677423910337928889088513649, 18.73254967837151298845443081013, 19.63211447989981842080958145107, 19.99884957419095034737950630088, 20.79525785818801944666676298219

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