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Properties

Label 1-1183-1183.531-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.660 + 0.750i$
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  + (0.391 − 0.919i)2-s + (−0.948 + 0.316i)3-s + (−0.692 − 0.721i)4-s + (0.464 − 0.885i)5-s + (−0.0804 + 0.996i)6-s + (−0.935 + 0.354i)8-s + (0.799 − 0.600i)9-s + (−0.632 − 0.774i)10-s + (0.600 − 0.799i)11-s + (0.885 + 0.464i)12-s + (−0.160 + 0.987i)15-s + (−0.0402 + 0.999i)16-s + (−0.632 + 0.774i)17-s + (−0.239 − 0.970i)18-s + (−0.866 + 0.5i)19-s + (−0.960 + 0.278i)20-s + ⋯
L(s)  = 1  + (0.391 − 0.919i)2-s + (−0.948 + 0.316i)3-s + (−0.692 − 0.721i)4-s + (0.464 − 0.885i)5-s + (−0.0804 + 0.996i)6-s + (−0.935 + 0.354i)8-s + (0.799 − 0.600i)9-s + (−0.632 − 0.774i)10-s + (0.600 − 0.799i)11-s + (0.885 + 0.464i)12-s + (−0.160 + 0.987i)15-s + (−0.0402 + 0.999i)16-s + (−0.632 + 0.774i)17-s + (−0.239 − 0.970i)18-s + (−0.866 + 0.5i)19-s + (−0.960 + 0.278i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.660 + 0.750i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2617358562 - 0.5791543202i\)
\(L(\frac12)\) \(\approx\) \(-0.2617358562 - 0.5791543202i\)
\(L(1)\) \(\approx\) \(0.5901752949 - 0.5566116661i\)
\(L(1)\) \(\approx\) \(0.5901752949 - 0.5566116661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.391 - 0.919i)T \)
3 \( 1 + (-0.948 + 0.316i)T \)
5 \( 1 + (0.464 - 0.885i)T \)
11 \( 1 + (0.600 - 0.799i)T \)
17 \( 1 + (-0.632 + 0.774i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.919 - 0.391i)T \)
31 \( 1 + (-0.822 - 0.568i)T \)
37 \( 1 + (0.903 - 0.428i)T \)
41 \( 1 + (-0.316 - 0.948i)T \)
43 \( 1 + (-0.428 + 0.903i)T \)
47 \( 1 + (0.239 - 0.970i)T \)
53 \( 1 + (-0.354 - 0.935i)T \)
59 \( 1 + (0.999 - 0.0402i)T \)
61 \( 1 + (-0.987 + 0.160i)T \)
67 \( 1 + (0.721 + 0.692i)T \)
71 \( 1 + (-0.979 - 0.200i)T \)
73 \( 1 + (0.992 + 0.120i)T \)
79 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + (0.663 + 0.748i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.534 + 0.845i)T \)
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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

−21.95619937194729426126605699406, −21.48547071331279480617433043935, −20.22268885111777592704266613663, −19.03617195175348419251606467216, −18.30868866217269078066543427886, −17.70179426876623352317686114075, −17.16727335236394856993716155438, −16.433877951894245616026248111590, −15.37127400864335013503427684813, −14.93264686105185709918098451556, −13.92647697520196274519934578103, −13.2229719053294126626714015291, −12.5294738914162350790664238002, −11.52958821447533997845228102914, −10.90439481329491903831558538820, −9.73684019617419097904192595131, −9.101231912822100257432775312200, −7.685194251538484895752040668018, −6.97898123987496664087133875459, −6.59126852359184914608967536273, −5.679498316505062256570258082750, −4.88096257507896422256362195117, −4.01977847502847866815734108172, −2.787323826778658491488844754979, −1.5973267271263087552125123294, 0.26232370501221071898589994079, 1.33376982429684016026678817953, 2.219028917861866713519417619712, 3.81318495003183681072677952845, 4.22874123471177848347628293438, 5.26200398430496624909255581300, 5.90660443470806168387436917382, 6.58160264549035493289349290404, 8.34127275028335881498454699315, 9.06037542841510437426332324145, 9.80601159893519466834646464355, 10.70090444452355362099088929965, 11.26401952272588148445699632605, 12.11144638116461925381495675430, 12.894841188445068070294841027198, 13.271377302909814244170507796199, 14.4949327414053497593355438804, 15.16430375179688028171554381876, 16.38858056611982900628575557252, 16.89687589112480261227615253009, 17.61393827079321613391437024199, 18.51400285028329387718250819891, 19.24447571265433049042411800228, 20.18465650189627482061534486287, 20.898621038859124331567803151235

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