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Properties

Label 1-260-260.227-r1-0-0
Degree $1$
Conductor $260$
Sign $0.507 + 0.861i$
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  + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s + i·21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + (−0.866 + 0.5i)41-s + (0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s + i·21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + (−0.866 + 0.5i)41-s + (0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.507 + 0.861i$
Analytic conductor: \(27.9408\)
Root analytic conductor: \(27.9408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (1:\ ),\ 0.507 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.822758708 + 1.041899411i\)
\(L(\frac12)\) \(\approx\) \(1.822758708 + 1.041899411i\)
\(L(1)\) \(\approx\) \(1.312810804 + 0.1312926369i\)
\(L(1)\) \(\approx\) \(1.312810804 + 0.1312926369i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + T \)
53 \( 1 - iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−25.75854807289780962058229277133, −24.68783112653183236055526368392, −23.74692921812292220147180102911, −22.7074999645223900067723451944, −21.75524911904627252097887331839, −20.72746334619556489974935915217, −20.18806815173830763044901536728, −19.16698664354184127718295338909, −18.3456502801372696445070517136, −16.901814151686133411270926231353, −16.08334618624184817863400168235, −15.36634406328652819781028537285, −13.90547121117239172820511183602, −13.72479566428842979748464616618, −12.42370290163529432160450438350, −10.94723610940759706655452771435, −10.09773417040597860770460700691, −9.31318671453022153201598954318, −7.99407289381297991648867681743, −7.35156922749827283992258296057, −5.77821712462214185087862184479, −4.483167533990135843535419873102, −3.44140664429480418492133401363, −2.46719053514251435929966550265, −0.60087705055952565328322565218, 1.42268606674504629986636616148, 2.65682201853045912544269988971, 3.53886124626226544077745287818, 5.20230907074784218563422733588, 6.341908378475445631591330873497, 7.56515215377453961376923581436, 8.34278082406918272749321932905, 9.47159333224783574392736922190, 10.25274582633549496868766473834, 12.013548941639456545658188983372, 12.553846007747722091441037467961, 13.58473234802728342005203243606, 14.56793760519405529815636741828, 15.464248701606341753341851827003, 16.27540252963815161535014430299, 17.88506912559648435351076964497, 18.457631948575509011702718271188, 19.34774583802021232037598951193, 20.23167592092980421483689462168, 21.14421100312114675589791487467, 22.04017089853320294222504821668, 23.290965747288920591652626400624, 24.02266122005873726904885112613, 25.206168569763524261480117580739, 25.583928323700287961963532751816

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