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Properties

Label 1-2183-2183.1084-r0-0-0
Degree $1$
Conductor $2183$
Sign $-0.275 + 0.961i$
Analytic cond. $10.1378$
Root an. cond. $10.1378$
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  + (−0.197 + 0.980i)2-s + (0.874 + 0.484i)3-s + (−0.922 − 0.386i)4-s + (0.619 − 0.785i)5-s + (−0.647 + 0.762i)6-s + (−0.0901 − 0.995i)7-s + (0.561 − 0.827i)8-s + (0.530 + 0.847i)9-s + (0.647 + 0.762i)10-s + (0.267 + 0.963i)11-s + (−0.619 − 0.785i)12-s + (−0.530 + 0.847i)13-s + (0.994 + 0.108i)14-s + (0.922 − 0.386i)15-s + (0.700 + 0.713i)16-s + (0.0901 − 0.995i)17-s + ⋯
L(s)  = 1  + (−0.197 + 0.980i)2-s + (0.874 + 0.484i)3-s + (−0.922 − 0.386i)4-s + (0.619 − 0.785i)5-s + (−0.647 + 0.762i)6-s + (−0.0901 − 0.995i)7-s + (0.561 − 0.827i)8-s + (0.530 + 0.847i)9-s + (0.647 + 0.762i)10-s + (0.267 + 0.963i)11-s + (−0.619 − 0.785i)12-s + (−0.530 + 0.847i)13-s + (0.994 + 0.108i)14-s + (0.922 − 0.386i)15-s + (0.700 + 0.713i)16-s + (0.0901 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $-0.275 + 0.961i$
Analytic conductor: \(10.1378\)
Root analytic conductor: \(10.1378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (0:\ ),\ -0.275 + 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.236265122 + 1.640725072i\)
\(L(\frac12)\) \(\approx\) \(1.236265122 + 1.640725072i\)
\(L(1)\) \(\approx\) \(1.136554044 + 0.6899820000i\)
\(L(1)\) \(\approx\) \(1.136554044 + 0.6899820000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.197 + 0.980i)T \)
3 \( 1 + (0.874 + 0.484i)T \)
5 \( 1 + (0.619 - 0.785i)T \)
7 \( 1 + (-0.0901 - 0.995i)T \)
11 \( 1 + (0.267 + 0.963i)T \)
13 \( 1 + (-0.530 + 0.847i)T \)
17 \( 1 + (0.0901 - 0.995i)T \)
19 \( 1 + (0.302 + 0.953i)T \)
23 \( 1 + (0.161 + 0.986i)T \)
29 \( 1 + (0.947 + 0.319i)T \)
31 \( 1 + (-0.976 + 0.214i)T \)
41 \( 1 + (0.935 - 0.353i)T \)
43 \( 1 + (-0.267 + 0.963i)T \)
47 \( 1 + (-0.370 + 0.928i)T \)
53 \( 1 + (0.336 + 0.941i)T \)
61 \( 1 + (-0.750 - 0.661i)T \)
67 \( 1 + (0.997 + 0.0721i)T \)
71 \( 1 + (0.989 - 0.143i)T \)
73 \( 1 + (-0.994 - 0.108i)T \)
79 \( 1 + (0.0180 - 0.999i)T \)
83 \( 1 + (0.837 + 0.546i)T \)
89 \( 1 + (-0.750 + 0.661i)T \)
97 \( 1 + (0.994 - 0.108i)T \)
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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

−19.62188974819583618723989947561, −18.78312605545555924836413405532, −18.41579450088207335060331969589, −17.73186491019963964649231439681, −17.032528045147286075373164238, −15.75331982913852831788824640141, −14.776790129062116075829392498755, −14.496186893331839462781882555912, −13.51931713140461115233947945307, −13.01269353713525078281577682881, −12.31568755754562977412996073748, −11.497202978832181421330181892607, −10.637996318648092457799962464151, −9.96199035187609050207362596159, −9.15543742431772558879201761401, −8.59110640898056362103305128854, −7.92043860287939744279509501770, −6.84798328930128689294864175197, −5.989101122364544193928665806060, −5.14451959305974445676469298119, −3.81743586204952354012918778949, −3.06076773897582865361015041044, −2.54717078934544084319738202946, −1.90120925216029789397928329553, −0.70752017279184483616513483715, 1.16746394614408101809774660321, 1.89502542091707123337997532316, 3.30722372656914153011507390255, 4.36129663435597544771824362471, 4.64880489398625791839994177094, 5.53404686116622076467461234850, 6.70730514283601052719131785349, 7.402119587653386582503715211280, 7.93350490773665127218368660385, 8.99870684073551288951964629760, 9.65912373876465917909627392152, 9.749676618711929375366210253777, 10.81523983911905824781479573057, 12.23612060887934715060982576170, 12.97446328307859166171423130959, 13.84004865659134280965046471574, 14.14523069936677932978956684986, 14.76223728093829693871761651242, 15.863445025419375099237789043304, 16.30050545432226382750816195939, 16.91441552697098894066744493271, 17.606411840657689135201675664506, 18.356115871594589558824978468016, 19.41463047324536601213598099073, 19.91534386571911788962791233915

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