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Properties

Label 1-1183-1183.902-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.778 + 0.627i$
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  + (0.239 + 0.970i)2-s + (−0.120 − 0.992i)3-s + (−0.885 + 0.464i)4-s + (−0.822 + 0.568i)5-s + (0.935 − 0.354i)6-s + (−0.663 − 0.748i)8-s + (−0.970 + 0.239i)9-s + (−0.748 − 0.663i)10-s + (0.239 − 0.970i)11-s + (0.568 + 0.822i)12-s + (0.663 + 0.748i)15-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s + (−0.464 − 0.885i)18-s + i·19-s + (0.464 − 0.885i)20-s + ⋯
L(s)  = 1  + (0.239 + 0.970i)2-s + (−0.120 − 0.992i)3-s + (−0.885 + 0.464i)4-s + (−0.822 + 0.568i)5-s + (0.935 − 0.354i)6-s + (−0.663 − 0.748i)8-s + (−0.970 + 0.239i)9-s + (−0.748 − 0.663i)10-s + (0.239 − 0.970i)11-s + (0.568 + 0.822i)12-s + (0.663 + 0.748i)15-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s + (−0.464 − 0.885i)18-s + i·19-s + (0.464 − 0.885i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (902, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8615932001 + 0.3040551061i\)
\(L(\frac12)\) \(\approx\) \(0.8615932001 + 0.3040551061i\)
\(L(1)\) \(\approx\) \(0.7835251668 + 0.2176239152i\)
\(L(1)\) \(\approx\) \(0.7835251668 + 0.2176239152i\)

Euler product

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

−20.83002224356638761915967332882, −20.498508971897455849451115889146, −19.84846077576471469504192040313, −19.24057701014949780543660916043, −17.9778633261014831943397103109, −17.44045388357838788193847692345, −16.43429073043609529846040943430, −15.53753584326171362709184385039, −15.074073512789960245789388645836, −14.128382096217317481829338089, −13.18837251519545246155386265301, −12.34435933772011150893846579413, −11.52285313399878052280591208743, −11.192733643710974106422575006729, −10.038757937136088990080964486200, −9.44635150135892087885169475598, −8.75731385943801320241896017414, −7.82720300468953062909646615524, −6.47838733841437367408243039012, −5.22148625171852382401428988722, −4.58049014544870610312558626965, −4.10506775190043738396322275079, −3.10790272753247583313343908195, −2.10756822527612364839386807154, −0.62599860095499099219473051996, 0.629617431614400652034845911005, 2.20265824210074927555563907680, 3.45834112690125191403807108163, 4.045009731028607409181938325620, 5.425964057893820799974020624131, 6.22988882175290154820582874797, 6.73057248561636314061089982808, 7.78615480496666482263545035165, 8.1454304210451982644579389164, 8.964175182778653860615267794980, 10.33346562515969150021449740388, 11.3641017948100543036551123389, 12.02873850688362189849217594016, 12.84245153225369058576943711033, 13.69252458800673533957621063544, 14.29998669443422781495882603546, 15.04509651257056148531310630579, 15.90166783668049915149616776520, 16.703599615761029658610310677500, 17.35018984870920681438622667013, 18.40853985965509005427781438949, 18.69659931787363568798632884311, 19.49147983024426129944122390649, 20.36541682683486879955120123752, 21.73452252364179724863317514101

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