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Properties

Label 1-260-260.99-r0-0-0
Degree $1$
Conductor $260$
Sign $0.957 - 0.289i$
Analytic cond. $1.20743$
Root an. cond. $1.20743$
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  + 3-s − i·7-s + 9-s + i·11-s + 17-s − i·19-s − i·21-s − 23-s + 27-s + 29-s − i·31-s + i·33-s + i·37-s + i·41-s − 43-s + ⋯
L(s)  = 1  + 3-s − i·7-s + 9-s + i·11-s + 17-s − i·19-s − i·21-s − 23-s + 27-s + 29-s − i·31-s + i·33-s + i·37-s + i·41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(1.20743\)
Root analytic conductor: \(1.20743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (0:\ ),\ 0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.730429216 - 0.2562224722i\)
\(L(\frac12)\) \(\approx\) \(1.730429216 - 0.2562224722i\)
\(L(1)\) \(\approx\) \(1.461201913 - 0.1152893140i\)
\(L(1)\) \(\approx\) \(1.461201913 - 0.1152893140i\)

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 \( 1 + T \)
11 \( 1 \)
17 \( 1 \)
19 \( 1 - iT \)
23 \( 1 \)
29 \( 1 + T \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + T \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 \)
89 \( 1 - T \)
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

−25.75658885457157733995996431957, −25.09969413767391686074537940955, −24.37574910125533748996096158184, −23.328366700871217857245632005378, −21.94614368186877567744269541442, −21.37045573714700689963224753194, −20.50109526156505678691902370210, −19.29132138910339587447064809411, −18.831164317199938904067063802043, −17.86176376043128410465650785149, −16.28774297562279840789075927161, −15.73802006925082745816952532177, −14.45861915999255341516343638065, −14.04118314020166137216139611953, −12.68997296311720898572064725435, −11.95542524336087794000403600781, −10.49558744588937231932889293121, −9.500130374826743073481850043063, −8.4915569654790296561971817463, −7.88265502323448648307999904960, −6.37489348892406344028768663179, −5.30152645854018239744286724456, −3.75032816681100594536096296914, −2.86308905055797860038870139012, −1.607310975278382856996896353604, 1.36671696086356107444359977476, 2.709708891466121093443906492839, 3.91148632757990961782404402322, 4.80784258096435079035516693676, 6.633476654506243100481078787601, 7.51465388455284221387707377641, 8.353110532113709841865093119940, 9.76816664382683194773357013722, 10.164551931589027271225134627049, 11.69509061607200102874723387117, 12.89111997779055162457038748286, 13.68666474133896265010866290099, 14.55797192973919966484957350330, 15.416933455688368138851403154127, 16.49996417453982246033889356101, 17.57091705252045364036742690675, 18.55523503931914450223826839212, 19.71169658770762921657780071353, 20.20060730662182246204504330309, 21.04673410701722246088244186988, 22.10537055973051645928764209088, 23.31501575821532282983899321838, 23.98667906487184116362680253433, 25.15584234273126159771170177882, 25.87145019383574743864357123743

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