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Properties

Label 1-1183-1183.1034-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.139 + 0.990i$
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  + (−0.0804 − 0.996i)2-s + (−0.885 − 0.464i)3-s + (−0.987 + 0.160i)4-s + (0.316 − 0.948i)5-s + (−0.391 + 0.919i)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 + (0.948 + 0.316i)12-s + (−0.721 + 0.692i)15-s + (0.948 − 0.316i)16-s + (0.692 + 0.721i)17-s + (0.774 − 0.632i)18-s − i·19-s + (−0.160 + 0.987i)20-s + ⋯
L(s)  = 1  + (−0.0804 − 0.996i)2-s + (−0.885 − 0.464i)3-s + (−0.987 + 0.160i)4-s + (0.316 − 0.948i)5-s + (−0.391 + 0.919i)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 + (0.948 + 0.316i)12-s + (−0.721 + 0.692i)15-s + (0.948 − 0.316i)16-s + (0.692 + 0.721i)17-s + (0.774 − 0.632i)18-s − i·19-s + (−0.160 + 0.987i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1034, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.139 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2771323537 - 0.3188268318i\)
\(L(\frac12)\) \(\approx\) \(-0.2771323537 - 0.3188268318i\)
\(L(1)\) \(\approx\) \(0.3892587660 - 0.4914436819i\)
\(L(1)\) \(\approx\) \(0.3892587660 - 0.4914436819i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.0804 - 0.996i)T \)
3 \( 1 + (-0.885 - 0.464i)T \)
5 \( 1 + (0.316 - 0.948i)T \)
11 \( 1 + (-0.822 - 0.568i)T \)
17 \( 1 + (0.692 + 0.721i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.428 - 0.903i)T \)
31 \( 1 + (-0.391 + 0.919i)T \)
37 \( 1 + (-0.391 + 0.919i)T \)
41 \( 1 + (-0.999 + 0.0402i)T \)
43 \( 1 + (-0.799 - 0.600i)T \)
47 \( 1 + (-0.774 - 0.632i)T \)
53 \( 1 + (0.278 - 0.960i)T \)
59 \( 1 + (-0.316 + 0.948i)T \)
61 \( 1 + (0.970 + 0.239i)T \)
67 \( 1 + (0.935 - 0.354i)T \)
71 \( 1 + (-0.534 - 0.845i)T \)
73 \( 1 + (-0.903 + 0.428i)T \)
79 \( 1 + (-0.632 + 0.774i)T \)
83 \( 1 + (-0.464 - 0.885i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.316 - 0.948i)T \)
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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

−21.9212874500925042182809408423, −21.36234627015448067552202088458, −20.431152685732767384741241306349, −18.940304674723232889680763949408, −18.47678167339325942368608400414, −17.810107141393525690221394550139, −17.189935000767466152478713777036, −16.29618333962422833575886242996, −15.71762757695873354351876949989, −14.89370652080575634926305137732, −14.359815609173922978256554383727, −13.321386649454463756937186151790, −12.51021709367078539528151390828, −11.49082002064574936461851549515, −10.52045713054520259360750915401, −9.93723201868323378736510361359, −9.333464702489446450924797464631, −7.93783763208239214919446485603, −7.21653094744184394304162350831, −6.54200732306683919813565491532, −5.53290903335431184488628101311, −5.19007981938640357538341329784, −3.98559647666476734566543726676, −3.09358650576071728352836404601, −1.46287178183586201715646623398, 0.22551345785445883503595735840, 1.18078736328572559152048847813, 2.07051071427456965316590311995, 3.18129350399537302905385351954, 4.492569760888287500922746621622, 5.12032574284317361358550631259, 5.78028395677273838811992571202, 6.96296641173557356629963928214, 8.294540014070494646162895629713, 8.56454830807102131643485750243, 9.96304329724423447560562904247, 10.37949131863872417383219876276, 11.37044020395671887836963201476, 12.006823859839779845994866297933, 12.84437905788938221697195844311, 13.23240607470399748135384973110, 13.97762065447098952089139580611, 15.336569444717422924981084647806, 16.43274740542612237956971134902, 16.95598669770214538087587564080, 17.67686358735797904638903017126, 18.405326397123531095015934591756, 19.11622498629889531027961488898, 19.84045001405773025937654075163, 20.80201803230104247143996288214

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