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Properties

Label 1-1183-1183.496-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.342 + 0.939i$
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  + (0.534 + 0.845i)2-s + (−0.692 + 0.721i)3-s + (−0.428 + 0.903i)4-s + (0.935 − 0.354i)5-s + (−0.979 − 0.200i)6-s + (−0.992 + 0.120i)8-s + (−0.0402 − 0.999i)9-s + (0.799 + 0.600i)10-s + (−0.999 − 0.0402i)11-s + (−0.354 − 0.935i)12-s + (−0.391 + 0.919i)15-s + (−0.632 − 0.774i)16-s + (0.799 − 0.600i)17-s + (0.822 − 0.568i)18-s + (0.866 − 0.5i)19-s + (−0.0804 + 0.996i)20-s + ⋯
L(s)  = 1  + (0.534 + 0.845i)2-s + (−0.692 + 0.721i)3-s + (−0.428 + 0.903i)4-s + (0.935 − 0.354i)5-s + (−0.979 − 0.200i)6-s + (−0.992 + 0.120i)8-s + (−0.0402 − 0.999i)9-s + (0.799 + 0.600i)10-s + (−0.999 − 0.0402i)11-s + (−0.354 − 0.935i)12-s + (−0.391 + 0.919i)15-s + (−0.632 − 0.774i)16-s + (0.799 − 0.600i)17-s + (0.822 − 0.568i)18-s + (0.866 − 0.5i)19-s + (−0.0804 + 0.996i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.363360578 + 0.9535865716i\)
\(L(\frac12)\) \(\approx\) \(1.363360578 + 0.9535865716i\)
\(L(1)\) \(\approx\) \(1.034723916 + 0.6362448787i\)
\(L(1)\) \(\approx\) \(1.034723916 + 0.6362448787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.534 + 0.845i)T \)
3 \( 1 + (-0.692 + 0.721i)T \)
5 \( 1 + (0.935 - 0.354i)T \)
11 \( 1 + (-0.999 - 0.0402i)T \)
17 \( 1 + (0.799 - 0.600i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.845 + 0.534i)T \)
31 \( 1 + (0.663 - 0.748i)T \)
37 \( 1 + (0.316 + 0.948i)T \)
41 \( 1 + (0.721 + 0.692i)T \)
43 \( 1 + (-0.948 - 0.316i)T \)
47 \( 1 + (-0.822 - 0.568i)T \)
53 \( 1 + (0.120 + 0.992i)T \)
59 \( 1 + (0.774 + 0.632i)T \)
61 \( 1 + (0.919 - 0.391i)T \)
67 \( 1 + (0.903 - 0.428i)T \)
71 \( 1 + (0.960 - 0.278i)T \)
73 \( 1 + (-0.464 - 0.885i)T \)
79 \( 1 + (0.568 - 0.822i)T \)
83 \( 1 + (-0.239 - 0.970i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.160 + 0.987i)T \)
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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

−21.2497256360388139746790118155, −20.50330148443259534930538085103, −19.303186530978637798135285651788, −18.8810804515477023016146174632, −17.991140269164531615800059690604, −17.659664598356520213341715502753, −16.5794592531532360081772505859, −15.57699262166677876316842582302, −14.467669732918820983225566215463, −13.89815138101157551555234354571, −12.98251129146889679269711036121, −12.74350670194313945195631130484, −11.60261322423668783899945661161, −11.01342702374860541406295984739, −10.12942918932084958103237825448, −9.67682948398927411377174657028, −8.28697902420284630966879825255, −7.280449153137225784491719347690, −6.30035640959976173452635597155, −5.394318984685789581549054989077, −5.25634940089579364202984075468, −3.674507546681451536162378866321, −2.67331113375160544421705346831, −1.86548895106552697523825589519, −1.004676619015234639244875535859, 0.764098570681962867853753078891, 2.59422214843359965255104227557, 3.43698236806239016814199597401, 4.81952006017108919437375582363, 5.05021682600473510922853898838, 5.87461194982752973535981850430, 6.63786701904092021923405143390, 7.65005341085250628483654540476, 8.68653314045839434827129466036, 9.54026372115471873691984324805, 10.13569804139082938038856552839, 11.25539480724891522104486069582, 12.10683672127014616616907488642, 12.98786487814523086507470611534, 13.56739177030343929836782617583, 14.54850730181860348966531383586, 15.219572016815523080661253601226, 16.147828380203789765147071401305, 16.583279619392421795078324245995, 17.249919521858706073274321946180, 18.15780482567665476374404998808, 18.49950967824464101085221204697, 20.4662148293107667642918661417, 20.74610538024235593146783418739, 21.56697646072202603530116365607

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