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Properties

Label 1-1183-1183.216-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.703 + 0.710i$
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  + (−0.822 + 0.568i)2-s + (−0.885 − 0.464i)3-s + (0.354 − 0.935i)4-s + (0.663 + 0.748i)5-s + (0.992 − 0.120i)6-s + (0.239 + 0.970i)8-s + (0.568 + 0.822i)9-s + (−0.970 − 0.239i)10-s + (−0.822 − 0.568i)11-s + (−0.748 + 0.663i)12-s + (−0.239 − 0.970i)15-s + (−0.748 − 0.663i)16-s + (−0.970 + 0.239i)17-s + (−0.935 − 0.354i)18-s − i·19-s + (0.935 − 0.354i)20-s + ⋯
L(s)  = 1  + (−0.822 + 0.568i)2-s + (−0.885 − 0.464i)3-s + (0.354 − 0.935i)4-s + (0.663 + 0.748i)5-s + (0.992 − 0.120i)6-s + (0.239 + 0.970i)8-s + (0.568 + 0.822i)9-s + (−0.970 − 0.239i)10-s + (−0.822 − 0.568i)11-s + (−0.748 + 0.663i)12-s + (−0.239 − 0.970i)15-s + (−0.748 − 0.663i)16-s + (−0.970 + 0.239i)17-s + (−0.935 − 0.354i)18-s − i·19-s + (0.935 − 0.354i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.703 + 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6215693965 + 0.2591562310i\)
\(L(\frac12)\) \(\approx\) \(0.6215693965 + 0.2591562310i\)
\(L(1)\) \(\approx\) \(0.5774619238 + 0.1172914676i\)
\(L(1)\) \(\approx\) \(0.5774619238 + 0.1172914676i\)

Euler product

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

−20.93490512054542181336223793398, −20.57263403383907232110272229586, −19.72582581613498298657072824052, −18.50894870491677953408006911368, −17.97741476420038510634370253469, −17.37190499188810261603947270355, −16.713867418764551234259124228178, −15.925164952427282812497716206710, −15.430399158561405514647709374895, −13.90566097607139387690940358441, −12.96685489792278741673631873867, −12.362290408244096016606848911416, −11.66701554579311988850911881048, −10.72299837914150828819865129038, −9.94731197507522395048158645502, −9.633838526806940430746076266725, −8.53715871361904580229862999175, −7.75211215615903814856688257875, −6.55730087785373775377432314431, −5.82850280526924958711573166750, −4.68288088678076895765561345233, −4.11072415183604380971953611293, −2.63812721774720746337215384034, −1.73694787905348351927761491492, −0.599803387790132604927118461670, 0.74783136346818876172416716250, 2.00926671432272568487562219432, 2.713597638186116569588971093278, 4.573337446406897723849521746886, 5.49212818928658219907152900933, 6.21325089676986796227774550453, 6.76024872531667333313982802636, 7.60777448025008180783644492707, 8.45651099901938470121466094249, 9.53888369006003623772656005577, 10.38126251911163746646564771229, 10.93193613343384650861221615742, 11.52619912029015845012607673269, 12.84565129861274796607477093303, 13.63648923358151204950506630708, 14.30189843794014495487208460435, 15.5409225614081669564814661603, 15.900971328883448621822481029406, 16.92882197863691018673701621996, 17.689697004522968671808047390987, 17.99357321213926349968304033832, 18.760799904003640534725751414520, 19.40209888056372190504373533131, 20.3613533899447853155884463108, 21.622203826511232475824626951738

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