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Properties

Label 1-3200-3200.323-r0-0-0
Degree $1$
Conductor $3200$
Sign $0.384 + 0.923i$
Analytic cond. $14.8607$
Root an. cond. $14.8607$
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  + (0.678 + 0.734i)3-s + (0.382 + 0.923i)7-s + (−0.0784 + 0.996i)9-s + (0.117 − 0.993i)11-s + (−0.619 + 0.785i)13-s + (0.156 − 0.987i)17-s + (0.938 + 0.346i)19-s + (−0.418 + 0.908i)21-s + (0.649 − 0.760i)23-s + (−0.785 + 0.619i)27-s + (0.678 + 0.734i)29-s + (−0.587 − 0.809i)31-s + (0.809 − 0.587i)33-s + (0.271 − 0.962i)37-s + (−0.996 + 0.0784i)39-s + ⋯
L(s)  = 1  + (0.678 + 0.734i)3-s + (0.382 + 0.923i)7-s + (−0.0784 + 0.996i)9-s + (0.117 − 0.993i)11-s + (−0.619 + 0.785i)13-s + (0.156 − 0.987i)17-s + (0.938 + 0.346i)19-s + (−0.418 + 0.908i)21-s + (0.649 − 0.760i)23-s + (−0.785 + 0.619i)27-s + (0.678 + 0.734i)29-s + (−0.587 − 0.809i)31-s + (0.809 − 0.587i)33-s + (0.271 − 0.962i)37-s + (−0.996 + 0.0784i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.384 + 0.923i$
Analytic conductor: \(14.8607\)
Root analytic conductor: \(14.8607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3200,\ (0:\ ),\ 0.384 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.063889511 + 1.376117952i\)
\(L(\frac12)\) \(\approx\) \(2.063889511 + 1.376117952i\)
\(L(1)\) \(\approx\) \(1.382520013 + 0.4852067565i\)
\(L(1)\) \(\approx\) \(1.382520013 + 0.4852067565i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.678 + 0.734i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (0.117 - 0.993i)T \)
13 \( 1 + (-0.619 + 0.785i)T \)
17 \( 1 + (0.156 - 0.987i)T \)
19 \( 1 + (0.938 + 0.346i)T \)
23 \( 1 + (0.649 - 0.760i)T \)
29 \( 1 + (0.678 + 0.734i)T \)
31 \( 1 + (-0.587 - 0.809i)T \)
37 \( 1 + (0.271 - 0.962i)T \)
41 \( 1 + (0.996 + 0.0784i)T \)
43 \( 1 + (0.980 - 0.195i)T \)
47 \( 1 + (0.987 - 0.156i)T \)
53 \( 1 + (0.418 - 0.908i)T \)
59 \( 1 + (-0.271 + 0.962i)T \)
61 \( 1 + (0.872 + 0.488i)T \)
67 \( 1 + (0.908 - 0.418i)T \)
71 \( 1 + (-0.972 - 0.233i)T \)
73 \( 1 + (0.760 + 0.649i)T \)
79 \( 1 + (-0.987 + 0.156i)T \)
83 \( 1 + (-0.346 + 0.938i)T \)
89 \( 1 + (0.996 - 0.0784i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−18.94809530492626742137139461567, −17.93570033266486056054347385965, −17.45789628288772402352775092273, −17.13968303478313168474714734805, −15.82495494692736643467770075264, −15.19684308205508997401885645441, −14.508338967283187071451702938599, −13.98820710185042078416017938245, −13.158104858687920783334683318424, −12.64906134304927330191174329511, −11.95360003234931589151055870306, −11.06704522232139121413945628947, −10.17913952568393284034340575875, −9.62168808646870497105781416152, −8.75868068593228704128452123767, −7.79361785955789460776860829435, −7.49655310496630507696043466017, −6.85086790602364007050467731137, −5.870478088901641156679084107239, −4.90561264193959601225082451477, −4.095130614556822215439864772815, −3.262362909450459718435860463394, −2.46037097374614210792740333261, −1.494380232739981998689623405072, −0.841612449537234634537624946271, 0.973947631799745971300484097353, 2.32522493965307397750654045394, 2.67616308098092960327037473800, 3.63806857869233779733138275669, 4.468885582998307279312913072823, 5.27884582855714983530724670431, 5.780742958208406954809340362451, 7.02067010564958370378050272002, 7.725299494393659118917322764032, 8.59542226894195972030594765985, 9.14665001091961487685071976711, 9.579818696752376178530417724882, 10.62753518564153049287531113520, 11.29894737092609516945367278964, 11.91609380082820093521248921295, 12.773407576010708020809130338016, 13.76599812048503933551663915412, 14.35754641283196964356038956827, 14.67918732187224824319192758859, 15.69034506661201611034910645189, 16.23374881570472365529752241888, 16.66559351339676607817903909444, 17.74931862456941430994174576328, 18.61186066671931129059567710459, 18.97936034363734671013899311062

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