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Properties

Label 1-1183-1183.1097-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.277 - 0.960i$
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  + (0.960 − 0.278i)2-s + (−0.799 − 0.600i)3-s + (0.845 − 0.534i)4-s + (0.0804 − 0.996i)5-s + (−0.935 − 0.354i)6-s + (0.663 − 0.748i)8-s + (0.278 + 0.960i)9-s + (−0.200 − 0.979i)10-s + (0.960 + 0.278i)11-s + (−0.996 − 0.0804i)12-s + (−0.663 + 0.748i)15-s + (0.428 − 0.903i)16-s + (0.948 − 0.316i)17-s + (0.534 + 0.845i)18-s + (0.866 − 0.5i)19-s + (−0.464 − 0.885i)20-s + ⋯
L(s)  = 1  + (0.960 − 0.278i)2-s + (−0.799 − 0.600i)3-s + (0.845 − 0.534i)4-s + (0.0804 − 0.996i)5-s + (−0.935 − 0.354i)6-s + (0.663 − 0.748i)8-s + (0.278 + 0.960i)9-s + (−0.200 − 0.979i)10-s + (0.960 + 0.278i)11-s + (−0.996 − 0.0804i)12-s + (−0.663 + 0.748i)15-s + (0.428 − 0.903i)16-s + (0.948 − 0.316i)17-s + (0.534 + 0.845i)18-s + (0.866 − 0.5i)19-s + (−0.464 − 0.885i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.277 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.565849118 - 2.083166957i\)
\(L(\frac12)\) \(\approx\) \(1.565849118 - 2.083166957i\)
\(L(1)\) \(\approx\) \(1.447769240 - 0.9266889090i\)
\(L(1)\) \(\approx\) \(1.447769240 - 0.9266889090i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.960 - 0.278i)T \)
3 \( 1 + (-0.799 - 0.600i)T \)
5 \( 1 + (0.0804 - 0.996i)T \)
11 \( 1 + (0.960 + 0.278i)T \)
17 \( 1 + (0.948 - 0.316i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.970 - 0.239i)T \)
31 \( 1 + (0.774 - 0.632i)T \)
37 \( 1 + (0.160 + 0.987i)T \)
41 \( 1 + (0.992 + 0.120i)T \)
43 \( 1 + (0.354 + 0.935i)T \)
47 \( 1 + (-0.534 + 0.845i)T \)
53 \( 1 + (0.948 - 0.316i)T \)
59 \( 1 + (0.903 - 0.428i)T \)
61 \( 1 + (-0.948 - 0.316i)T \)
67 \( 1 + (0.999 + 0.0402i)T \)
71 \( 1 + (-0.992 - 0.120i)T \)
73 \( 1 + (-0.960 - 0.278i)T \)
79 \( 1 + (-0.845 - 0.534i)T \)
83 \( 1 + (-0.992 + 0.120i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.822 + 0.568i)T \)
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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

−21.56492326087686370367989787579, −21.11412653010651967679254028977, −20.15302737570609976801305668533, −19.151595633947367726048362501373, −18.252105930112415412061930052720, −17.34357470971097220597146462685, −16.65858060839544345680995728580, −16.04145914829035240461079085736, −15.09694728451073580085522925138, −14.51774748709453819824777576213, −13.98610878422825293135537167227, −12.74717636236458372994048879244, −11.97144759110385048359605759661, −11.39150400885269655183650972901, −10.61087491291557447628957534715, −9.92061418759850062404795033192, −8.73617621150405330814770145964, −7.39451884327405549809201977148, −6.83330096553300727856100732587, −5.88431368408483632144939177877, −5.49222470343350187456173488942, −4.19266742195007450088942462313, −3.63725832066709145623466461190, −2.77337712975850174555568501724, −1.338629480540684174946529737997, 1.0242256545502435117338081566, 1.47993202963834215936231481435, 2.77098345687169063550453071784, 4.01323283184538337826417607918, 4.81950511116802364480304609100, 5.52747253829646487297604249178, 6.20433766485674730972298668546, 7.214474367641106878608938513820, 7.889206319545679320771541681581, 9.37744293924402488828661294241, 9.955717416578393240986009018549, 11.39232293239913835578291901294, 11.60064404718272668690059870877, 12.41550404642682704756643447200, 13.11936872839353334987835324364, 13.6988182430626828368604475292, 14.61306171273244116707392931043, 15.662669889453739121225283272199, 16.39776913771813465253378510490, 17.0145320813506244719873331238, 17.77069106365896060743913149388, 18.949587711125628716237011032532, 19.52565335035829282153697768125, 20.3391607945446661704210360511, 21.0728312773656765917577963562

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