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Properties

Label 1-2183-2183.319-r0-0-0
Degree $1$
Conductor $2183$
Sign $-0.0650 - 0.997i$
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  + (−0.484 + 0.874i)2-s + (−0.989 − 0.143i)3-s + (−0.530 − 0.847i)4-s + (0.915 + 0.403i)5-s + (0.605 − 0.796i)6-s + (0.197 − 0.980i)7-s + (0.998 − 0.0541i)8-s + (0.958 + 0.284i)9-s + (−0.796 + 0.605i)10-s + (−0.561 + 0.827i)11-s + (0.403 + 0.915i)12-s + (−0.284 − 0.958i)13-s + (0.762 + 0.647i)14-s + (−0.847 − 0.530i)15-s + (−0.436 + 0.899i)16-s + (−0.980 + 0.197i)17-s + ⋯
L(s)  = 1  + (−0.484 + 0.874i)2-s + (−0.989 − 0.143i)3-s + (−0.530 − 0.847i)4-s + (0.915 + 0.403i)5-s + (0.605 − 0.796i)6-s + (0.197 − 0.980i)7-s + (0.998 − 0.0541i)8-s + (0.958 + 0.284i)9-s + (−0.796 + 0.605i)10-s + (−0.561 + 0.827i)11-s + (0.403 + 0.915i)12-s + (−0.284 − 0.958i)13-s + (0.762 + 0.647i)14-s + (−0.847 − 0.530i)15-s + (−0.436 + 0.899i)16-s + (−0.980 + 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0650 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0650 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $-0.0650 - 0.997i$
Analytic conductor: \(10.1378\)
Root analytic conductor: \(10.1378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (0:\ ),\ -0.0650 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2103750161 - 0.2245422002i\)
\(L(\frac12)\) \(\approx\) \(0.2103750161 - 0.2245422002i\)
\(L(1)\) \(\approx\) \(0.5623329153 + 0.1364998733i\)
\(L(1)\) \(\approx\) \(0.5623329153 + 0.1364998733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.484 + 0.874i)T \)
3 \( 1 + (-0.989 - 0.143i)T \)
5 \( 1 + (0.915 + 0.403i)T \)
7 \( 1 + (0.197 - 0.980i)T \)
11 \( 1 + (-0.561 + 0.827i)T \)
13 \( 1 + (-0.284 - 0.958i)T \)
17 \( 1 + (-0.980 + 0.197i)T \)
19 \( 1 + (-0.353 + 0.935i)T \)
23 \( 1 + (0.963 - 0.267i)T \)
29 \( 1 + (-0.515 + 0.856i)T \)
31 \( 1 + (-0.986 - 0.161i)T \)
41 \( 1 + (-0.700 - 0.713i)T \)
43 \( 1 + (-0.827 + 0.561i)T \)
47 \( 1 + (0.994 + 0.108i)T \)
53 \( 1 + (0.126 - 0.992i)T \)
61 \( 1 + (0.999 - 0.0180i)T \)
67 \( 1 + (-0.891 - 0.452i)T \)
71 \( 1 + (0.590 - 0.806i)T \)
73 \( 1 + (0.647 - 0.762i)T \)
79 \( 1 + (0.785 - 0.619i)T \)
83 \( 1 + (0.817 + 0.576i)T \)
89 \( 1 + (-0.999 - 0.0180i)T \)
97 \( 1 + (0.762 - 0.647i)T \)
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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

−19.97854781660664667344960596385, −18.84662359974392144570193419826, −18.61085050380556451228746450852, −17.79669676239446438464820896826, −17.20431202658084394394481621949, −16.64316628080448321393108087818, −15.87370482311416110513395629016, −15.00546172672493003034453344843, −13.69027440099771387008125695542, −13.213967346332701055471915377528, −12.54538519251771884797811905224, −11.659774764580085162542537871980, −11.20533998525510151771432090625, −10.54021332874380710785755938739, −9.52683635788981304568430856673, −9.10326302035260046083856458358, −8.45808368902029491756681051075, −7.15796933549327886414308648129, −6.37686329474808867147791244135, −5.33780795536466768396064433063, −4.94964994437860580678486890853, −3.98342349988619186110727287823, −2.63799629479996608531132564310, −2.07629637282472017185250437142, −1.0877551007907448528720060226, 0.153225576881133831110200795180, 1.39551403074387597531693548312, 2.1004780225556409820812555035, 3.73459163142704317744441639868, 4.870945569701387548679404437857, 5.243019945595972636606683175061, 6.13666273556071930415418098699, 6.933103811390048721709694236812, 7.292034126738329479507342173392, 8.19810693801662316520415390122, 9.36698366098919549877263724668, 10.09471851457325227903846767652, 10.6693996235284008908923352467, 10.95876735051559455362740232278, 12.54828531672230660534068958165, 13.10383705712903225261817255834, 13.700673718030050657193548944231, 14.844221580938722013469454917334, 15.09660374206986519909237948606, 16.2542563582526827635986949517, 16.87800011794898612533500384914, 17.37003084545785831236075707438, 18.008711691330533523616903400776, 18.33520626071481198987353369370, 19.31627132650035979386242951095

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