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Properties

Label 1-2864-2864.205-r1-0-0
Degree $1$
Conductor $2864$
Sign $0.995 - 0.0998i$
Analytic cond. $307.779$
Root an. cond. $307.779$
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  + (0.987 + 0.158i)3-s + (0.835 − 0.550i)5-s + (0.990 − 0.140i)7-s + (0.949 + 0.312i)9-s + (−0.932 + 0.362i)11-s + (0.815 − 0.579i)13-s + (0.911 − 0.411i)15-s + (0.0176 + 0.999i)17-s + (0.648 + 0.760i)19-s + (0.999 + 0.0176i)21-s + (−0.192 − 0.981i)23-s + (0.394 − 0.918i)25-s + (0.888 + 0.458i)27-s + (−0.996 + 0.0881i)29-s + (0.938 + 0.345i)31-s + ⋯
L(s)  = 1  + (0.987 + 0.158i)3-s + (0.835 − 0.550i)5-s + (0.990 − 0.140i)7-s + (0.949 + 0.312i)9-s + (−0.932 + 0.362i)11-s + (0.815 − 0.579i)13-s + (0.911 − 0.411i)15-s + (0.0176 + 0.999i)17-s + (0.648 + 0.760i)19-s + (0.999 + 0.0176i)21-s + (−0.192 − 0.981i)23-s + (0.394 − 0.918i)25-s + (0.888 + 0.458i)27-s + (−0.996 + 0.0881i)29-s + (0.938 + 0.345i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2864 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2864 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2864\)    =    \(2^{4} \cdot 179\)
Sign: $0.995 - 0.0998i$
Analytic conductor: \(307.779\)
Root analytic conductor: \(307.779\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2864} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2864,\ (1:\ ),\ 0.995 - 0.0998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.701283095 - 0.2853120509i\)
\(L(\frac12)\) \(\approx\) \(5.701283095 - 0.2853120509i\)
\(L(1)\) \(\approx\) \(2.114861252 - 0.05487388983i\)
\(L(1)\) \(\approx\) \(2.114861252 - 0.05487388983i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
179 \( 1 \)
good3 \( 1 + (0.987 + 0.158i)T \)
5 \( 1 + (0.835 - 0.550i)T \)
7 \( 1 + (0.990 - 0.140i)T \)
11 \( 1 + (-0.932 + 0.362i)T \)
13 \( 1 + (0.815 - 0.579i)T \)
17 \( 1 + (0.0176 + 0.999i)T \)
19 \( 1 + (0.648 + 0.760i)T \)
23 \( 1 + (-0.192 - 0.981i)T \)
29 \( 1 + (-0.996 + 0.0881i)T \)
31 \( 1 + (0.938 + 0.345i)T \)
37 \( 1 + (0.996 + 0.0881i)T \)
41 \( 1 + (-0.635 - 0.772i)T \)
43 \( 1 + (-0.918 + 0.394i)T \)
47 \( 1 + (-0.0529 + 0.998i)T \)
53 \( 1 + (0.888 - 0.458i)T \)
59 \( 1 + (0.725 + 0.688i)T \)
61 \( 1 + (0.105 - 0.994i)T \)
67 \( 1 + (0.378 - 0.925i)T \)
71 \( 1 + (-0.737 - 0.675i)T \)
73 \( 1 + (0.427 - 0.904i)T \)
79 \( 1 + (0.783 + 0.621i)T \)
83 \( 1 + (0.0352 - 0.999i)T \)
89 \( 1 + (-0.607 + 0.794i)T \)
97 \( 1 + (-0.227 + 0.973i)T \)
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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

−18.73635893881682967519306536525, −18.29686992104820814121262129251, −18.00973521995734481000735411999, −16.972330418654972386397688153353, −16.01218278941768662286840232563, −15.31693389073097397794117340183, −14.752074837894518675531141414939, −13.79711143192608362799141207202, −13.63501168052967185500154844738, −13.047378511033893056858169134189, −11.54732899738276722750775705447, −11.38033256528106272118190018971, −10.22579212240392915396593113635, −9.65519037772257402170025302203, −8.87998037162934368038824634919, −8.21706267851267829144620607444, −7.40396666716689191978202731671, −6.851399369486960626210254073821, −5.724127464252830196773700056287, −5.11407799480681494604779229154, −4.09805491415099700077140290767, −3.110291291107000869250090141102, −2.48939685403953966153453478602, −1.75475113805163339466120234130, −0.89881243104306321537072224851, 0.89268907794434087277100134978, 1.70045943537273097325728502681, 2.280568835481580755290530899876, 3.295916490855907153826849718497, 4.20296860294091236047042532698, 4.9569947793723302233303223560, 5.641432749268371228680871772053, 6.579439333371183031771186505182, 7.83671543251260626701042840935, 8.09636301749003625450123486013, 8.7529120341400695122747109866, 9.66364244202807808086594933355, 10.378400961218242051524238214088, 10.77438026662868034611578335601, 12.072066436168511054832532822776, 12.84956745298236695204258589045, 13.37566322260943508867228208667, 14.0018438887979477451552738491, 14.76171231878431993456085800323, 15.269920087710241139269357028213, 16.17854735101021833933970529877, 16.80400526261913047656031097484, 17.80167659736370352048545722678, 18.20887134092857630070859249566, 18.85724084357294568994994461932

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