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Properties

Label 1-1183-1183.563-r1-0-0
Degree $1$
Conductor $1183$
Sign $-0.475 + 0.879i$
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  + (0.970 + 0.239i)2-s + (−0.799 − 0.600i)3-s + (0.885 + 0.464i)4-s + (0.428 + 0.903i)5-s + (−0.632 − 0.774i)6-s + (0.748 + 0.663i)8-s + (0.278 + 0.960i)9-s + (0.200 + 0.979i)10-s + (−0.278 + 0.960i)11-s + (−0.428 − 0.903i)12-s + (0.200 − 0.979i)15-s + (0.568 + 0.822i)16-s + (0.748 + 0.663i)17-s + (0.0402 + 0.999i)18-s + (−0.5 − 0.866i)19-s + (−0.0402 + 0.999i)20-s + ⋯
L(s)  = 1  + (0.970 + 0.239i)2-s + (−0.799 − 0.600i)3-s + (0.885 + 0.464i)4-s + (0.428 + 0.903i)5-s + (−0.632 − 0.774i)6-s + (0.748 + 0.663i)8-s + (0.278 + 0.960i)9-s + (0.200 + 0.979i)10-s + (−0.278 + 0.960i)11-s + (−0.428 − 0.903i)12-s + (0.200 − 0.979i)15-s + (0.568 + 0.822i)16-s + (0.748 + 0.663i)17-s + (0.0402 + 0.999i)18-s + (−0.5 − 0.866i)19-s + (−0.0402 + 0.999i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.475 + 0.879i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ -0.475 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.753544148 + 2.940811273i\)
\(L(\frac12)\) \(\approx\) \(1.753544148 + 2.940811273i\)
\(L(1)\) \(\approx\) \(1.581039535 + 0.6435334527i\)
\(L(1)\) \(\approx\) \(1.581039535 + 0.6435334527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.970 + 0.239i)T \)
3 \( 1 + (-0.799 - 0.600i)T \)
5 \( 1 + (0.428 + 0.903i)T \)
11 \( 1 + (-0.278 + 0.960i)T \)
17 \( 1 + (0.748 + 0.663i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.278 + 0.960i)T \)
31 \( 1 + (0.987 - 0.160i)T \)
37 \( 1 + (0.354 - 0.935i)T \)
41 \( 1 + (-0.919 + 0.391i)T \)
43 \( 1 + (0.987 + 0.160i)T \)
47 \( 1 + (-0.0402 + 0.999i)T \)
53 \( 1 + (-0.200 + 0.979i)T \)
59 \( 1 + (0.568 - 0.822i)T \)
61 \( 1 + (-0.948 - 0.316i)T \)
67 \( 1 + (0.0402 - 0.999i)T \)
71 \( 1 + (-0.799 - 0.600i)T \)
73 \( 1 + (0.692 + 0.721i)T \)
79 \( 1 + (-0.0402 + 0.999i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.996 + 0.0804i)T \)
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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

−20.90895517635997121732970342251, −20.60020956890291725462194080600, −19.31236305055600247080493045780, −18.6158474582970486020500216203, −17.36269238314700777146684070349, −16.6835648205437456250684413380, −16.19535174726623586788854985602, −15.426599696839604459705460949690, −14.53495062119848720400477391399, −13.58706066196055180417283081103, −13.01898729455494770673454600875, −11.94147229790524485484760319662, −11.72935703160199831978615512718, −10.50196918216207241452943111214, −10.06688782955571408022439940515, −9.03359701916042666318476225695, −7.9670036412034797919816242737, −6.6532145145662439820405922435, −5.886310229730715692564518873587, −5.299402077778931487729822221154, −4.59962334438719534525346402011, −3.70420257376234822999214336132, −2.72433597457098324282836054719, −1.31836269571407576047960926319, −0.54186350240842621108625992637, 1.31323158311279605634637419748, 2.28750144266079050935825993005, 3.05067251130956811228168485957, 4.37402965956420212184479361299, 5.15061147192358426737926202275, 5.99913106687438401242387153004, 6.72397918440913171886608612374, 7.26328787467989834784274629307, 8.09560850435877003293294777496, 9.653560699635515950499730259806, 10.8176147493137315923849014924, 10.92390521525119744134072970877, 12.1741506666060603414257231922, 12.67361751509613530234562654276, 13.43550526058090585164522347457, 14.2388114176469572299800705319, 15.027813533486160345971765795606, 15.670811221497981082212815445285, 16.78402527112862134365603960303, 17.37980913381269381332041568335, 18.004478753765632945390096474008, 19.00763418653854353842759776010, 19.65777606067628822363290046724, 20.83515366417622924153834748627, 21.5712467253398351071999817257

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