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Properties

Label 1-1205-1205.1037-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.0515 - 0.998i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (−0.707 + 0.707i)2-s + (0.156 − 0.987i)3-s − i·4-s + (0.587 + 0.809i)6-s + (0.760 − 0.649i)7-s + (0.707 + 0.707i)8-s + (−0.951 − 0.309i)9-s + (0.923 − 0.382i)11-s + (−0.987 − 0.156i)12-s + (−0.233 − 0.972i)13-s + (−0.0784 + 0.996i)14-s − 16-s + (0.996 + 0.0784i)17-s + (0.891 − 0.453i)18-s + (−0.382 − 0.923i)19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.156 − 0.987i)3-s − i·4-s + (0.587 + 0.809i)6-s + (0.760 − 0.649i)7-s + (0.707 + 0.707i)8-s + (−0.951 − 0.309i)9-s + (0.923 − 0.382i)11-s + (−0.987 − 0.156i)12-s + (−0.233 − 0.972i)13-s + (−0.0784 + 0.996i)14-s − 16-s + (0.996 + 0.0784i)17-s + (0.891 − 0.453i)18-s + (−0.382 − 0.923i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.0515 - 0.998i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.0515 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8538771447 - 0.8991165947i\)
\(L(\frac12)\) \(\approx\) \(0.8538771447 - 0.8991165947i\)
\(L(1)\) \(\approx\) \(0.8708149446 - 0.2779370745i\)
\(L(1)\) \(\approx\) \(0.8708149446 - 0.2779370745i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.156 - 0.987i)T \)
7 \( 1 + (0.760 - 0.649i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (-0.233 - 0.972i)T \)
17 \( 1 + (0.996 + 0.0784i)T \)
19 \( 1 + (-0.382 - 0.923i)T \)
23 \( 1 + (0.852 + 0.522i)T \)
29 \( 1 + (-0.453 - 0.891i)T \)
31 \( 1 + (-0.0784 + 0.996i)T \)
37 \( 1 + (0.233 - 0.972i)T \)
41 \( 1 + (0.453 - 0.891i)T \)
43 \( 1 + (-0.649 + 0.760i)T \)
47 \( 1 + (0.891 - 0.453i)T \)
53 \( 1 + (0.453 + 0.891i)T \)
59 \( 1 + (-0.156 - 0.987i)T \)
61 \( 1 + (0.156 - 0.987i)T \)
67 \( 1 + (0.891 + 0.453i)T \)
71 \( 1 + (-0.649 + 0.760i)T \)
73 \( 1 + (-0.233 + 0.972i)T \)
79 \( 1 + (0.156 + 0.987i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.923 - 0.382i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−21.02517539194038014481702691697, −20.85217327730055674837306428576, −19.930352322730824450361675342375, −19.03394255732566759843245677141, −18.52333047512945218150747633763, −17.451857579974831983561857174675, −16.667399031995837126533832998197, −16.45023517249693280225225296555, −14.95456250631241144913767083182, −14.714176365795264400958825072597, −13.68358801492852112509501486936, −12.35737258404074098367551188390, −11.814657291324108721504443356751, −11.15918556476940393022332167824, −10.27453269879048032847439868980, −9.4962033239128553385596065644, −8.92698081376955230139446027150, −8.21639528851479096188897289832, −7.25042495530487154378980253017, −6.02045721748160579863359433250, −4.83437885719906274478665351911, −4.16645966298578953513265905116, −3.2649186085478120011606500781, −2.2339843456422326679399445000, −1.36892413366153439995858927859, 0.718364843037017826334862356670, 1.31944543509568202570247755337, 2.47895446184332858364602666950, 3.77198974423917503508890865777, 5.12861333058198244221333498087, 5.79999023638811619249973885009, 6.8534024620392861549713877344, 7.38684203233846271819519868827, 8.12602682090386458164839078043, 8.81442909177492713182106850283, 9.72239701003495904634908646593, 10.85899652821288685951269004818, 11.35159436971227925299268831045, 12.45161543863489582058029632980, 13.42717921424231606583888858825, 14.16261688074839375736484239232, 14.69249992338236096038950363364, 15.51320454345870997872643790753, 16.75435700600089987435098727788, 17.29709622053070171551979883575, 17.68383040763021065929514117846, 18.63514086331539922310209076460, 19.35540355445300532649758218349, 19.89434836324549838132823031151, 20.62762274720763387098980731944

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