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Properties

Label 1-260-260.83-r1-0-0
Degree $1$
Conductor $260$
Sign $0.256 + 0.966i$
Analytic cond. $27.9408$
Root an. cond. $27.9408$
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  − i·3-s − 7-s − 9-s − i·11-s + i·17-s − i·19-s + i·21-s + i·23-s + i·27-s − 29-s + i·31-s − 33-s + 37-s − i·41-s + i·43-s + ⋯
L(s)  = 1  − i·3-s − 7-s − 9-s − i·11-s + i·17-s − i·19-s + i·21-s + i·23-s + i·27-s − 29-s + i·31-s − 33-s + 37-s − i·41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.256 + 0.966i$
Analytic conductor: \(27.9408\)
Root analytic conductor: \(27.9408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (1:\ ),\ 0.256 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4054398590 + 0.3118226147i\)
\(L(\frac12)\) \(\approx\) \(0.4054398590 + 0.3118226147i\)
\(L(1)\) \(\approx\) \(0.7532252852 - 0.2000298488i\)
\(L(1)\) \(\approx\) \(0.7532252852 - 0.2000298488i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 \)
17 \( 1 \)
19 \( 1 - T \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 + iT \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

−25.60826569435650489433995771377, −24.83739295465349092652850947791, −23.263190507210647645455072356010, −22.642040096122393337520681799774, −22.01752912257398983660881706337, −20.62772882643711600109378880563, −20.33865305059036499167934825496, −19.09129000257841166656263878855, −18.05690851492317029413223081096, −16.75711372765336901834616516775, −16.27376443834304740097376426217, −15.2108878746913446084290978340, −14.46653171655395654820211206836, −13.19737232156118236215846646725, −12.177195126824324874950660455070, −11.08196171925004387602859659440, −9.79566855478933865569819500674, −9.60776711565450309740261198169, −8.18253692741828971004361260589, −6.85121112571692566540116203707, −5.71464267555651244338905571934, −4.55500271325078877596642601586, −3.55280816601168227470162632826, −2.37026760252154234653214757443, −0.172265469609425367360562419355, 1.158899915404866759701333510750, 2.66036022319062961477531739874, 3.66004198952386055150133297094, 5.536650501656135171677241753881, 6.346983433173707027007316696186, 7.30479724118214230361853201945, 8.44909474753992177307202315721, 9.37209550336481948426080208009, 10.7905674380200407084279169927, 11.70375224361985146276360266404, 12.91201301310202750303276073056, 13.32937283938456314964343438789, 14.43781595390516597278471146761, 15.66888303166841222380297641947, 16.68879041986803860056789460275, 17.55755622715303362838338514333, 18.631179005688829855986648319245, 19.41662887407576977936938630871, 19.917490799315540273403732621, 21.44473671018011723061992768597, 22.251808590249847627527924848437, 23.33058255172928072126302421590, 23.98239440508835589867032605797, 24.89535807356232202390833762020, 25.8535465002048766537612993166

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