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Properties

Label 1-1183-1183.17-r1-0-0
Degree $1$
Conductor $1183$
Sign $-0.436 + 0.899i$
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  + (−0.120 + 0.992i)2-s + (0.200 + 0.979i)3-s + (−0.970 − 0.239i)4-s + (−0.0402 + 0.999i)5-s + (−0.996 + 0.0804i)6-s + (0.354 − 0.935i)8-s + (−0.919 + 0.391i)9-s + (−0.987 − 0.160i)10-s + (0.919 + 0.391i)11-s + (0.0402 − 0.999i)12-s + (−0.987 + 0.160i)15-s + (0.885 + 0.464i)16-s + (0.354 − 0.935i)17-s + (−0.278 − 0.960i)18-s + (−0.5 − 0.866i)19-s + (0.278 − 0.960i)20-s + ⋯
L(s)  = 1  + (−0.120 + 0.992i)2-s + (0.200 + 0.979i)3-s + (−0.970 − 0.239i)4-s + (−0.0402 + 0.999i)5-s + (−0.996 + 0.0804i)6-s + (0.354 − 0.935i)8-s + (−0.919 + 0.391i)9-s + (−0.987 − 0.160i)10-s + (0.919 + 0.391i)11-s + (0.0402 − 0.999i)12-s + (−0.987 + 0.160i)15-s + (0.885 + 0.464i)16-s + (0.354 − 0.935i)17-s + (−0.278 − 0.960i)18-s + (−0.5 − 0.866i)19-s + (0.278 − 0.960i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.436 + 0.899i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ -0.436 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9900368244 + 1.581358134i\)
\(L(\frac12)\) \(\approx\) \(0.9900368244 + 1.581358134i\)
\(L(1)\) \(\approx\) \(0.6770116864 + 0.7704161719i\)
\(L(1)\) \(\approx\) \(0.6770116864 + 0.7704161719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.120 + 0.992i)T \)
3 \( 1 + (0.200 + 0.979i)T \)
5 \( 1 + (-0.0402 + 0.999i)T \)
11 \( 1 + (0.919 + 0.391i)T \)
17 \( 1 + (0.354 - 0.935i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.919 + 0.391i)T \)
31 \( 1 + (0.428 - 0.903i)T \)
37 \( 1 + (-0.568 - 0.822i)T \)
41 \( 1 + (0.948 + 0.316i)T \)
43 \( 1 + (0.428 + 0.903i)T \)
47 \( 1 + (0.278 - 0.960i)T \)
53 \( 1 + (0.987 - 0.160i)T \)
59 \( 1 + (0.885 - 0.464i)T \)
61 \( 1 + (0.632 - 0.774i)T \)
67 \( 1 + (-0.278 + 0.960i)T \)
71 \( 1 + (0.200 + 0.979i)T \)
73 \( 1 + (0.799 - 0.600i)T \)
79 \( 1 + (0.278 - 0.960i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.845 + 0.534i)T \)
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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

−20.85549511667810980169344651233, −19.80943113425399478617609158147, −19.314484287066008584874138629183, −18.84322943604910438790124738088, −17.75865989656499525843944529254, −17.029540918257253618835971027959, −16.69971484396831442806032875636, −15.109661980594537617517965405, −14.214309395505500933497210621774, −13.56564125638298604493814949661, −12.69428743202026445912480301846, −12.33263239317643741602733933164, −11.57953762113633777610043351022, −10.684955705065520771655717285518, −9.566612460056772342731819014813, −8.734475509983482391219382657825, −8.37390487584258183344605648298, −7.373138366705057863304330318643, −6.11560478014087124948370960695, −5.35357368943135031321615486074, −4.125011775792683183164156981907, −3.44870494410150422586919284503, −2.203284021407950167072084230981, −1.353796429872756941797911714266, −0.784750097902714843329297988727, 0.5521486749321807211960019059, 2.35259223096693944041553668617, 3.44668465683618054394478495935, 4.18901364563063084756686783984, 5.0784700706066106804674155902, 5.97726098903956956538631411333, 6.921799436700590494592075634231, 7.49843455135182022987002561024, 8.6765792426345962360196360938, 9.36684047506259622302328287621, 9.94723649352241668048944869401, 10.9365091089521232256152493240, 11.60168376770946548871223587737, 12.99981778213909862126422380009, 13.92382937131550550750319322839, 14.62887911070818960736742144756, 14.993310664583829776029576959892, 15.77420765045358028439419699203, 16.568236239993685502769459558463, 17.3139901263180535603725580052, 17.95539229025993727292861901778, 19.01102561936273875700170253480, 19.50992318814769777952733420354, 20.57591572551861876934879468127, 21.56713034123921745368352802681

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