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Properties

Label 1-260-260.123-r1-0-0
Degree $1$
Conductor $260$
Sign $-0.997 + 0.0685i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.997 + 0.0685i$
Analytic conductor: \(27.9408\)
Root analytic conductor: \(27.9408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (1:\ ),\ -0.997 + 0.0685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004207427921 + 0.1226365088i\)
\(L(\frac12)\) \(\approx\) \(0.004207427921 + 0.1226365088i\)
\(L(1)\) \(\approx\) \(0.6804912626 + 0.05790916179i\)
\(L(1)\) \(\approx\) \(0.6804912626 + 0.05790916179i\)

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

−24.8949004561768004303503571554, −24.28118860443805641203328478359, −23.53283481560649412867177429910, −22.426903846876099133993343928292, −21.71751588060184826331182370136, −20.83531760402451769154757761712, −19.42233750214969828648132005192, −18.58988159042348987705685807789, −17.87321817337968121727138318474, −17.0385467693921117899905760437, −15.83884445858944032900857361331, −15.17744796540189053386404349570, −13.61070320936629242679235635738, −12.97040065023594806723042971916, −11.7129173629440799784146752028, −11.24217859594445854742810196233, −10.04896621281552715077771422514, −8.646882441972722581688485980783, −7.71759305986180624091384675290, −6.50420606827826550293450653071, −5.50374510421639437205492276078, −4.72855635426230477246448289828, −2.84244282286766344873462724278, −1.59775412138456549109117639392, −0.04466024402214123933830411237, 1.38612288405027441381137652463, 3.23831090813916588161290333100, 4.627417421894356406337189371555, 5.16463343123205507968832893821, 6.66674585023040646844861738452, 7.49572333217700785533010418636, 8.93026063877433282194845168467, 10.20612435982475078252147301611, 10.744124930246820389311018710242, 11.77039700553173296579626690255, 12.81384120186594649259842978001, 13.918168842990144038587767200102, 15.059419425511254097817715682222, 16.003216670127981304089516098033, 16.791346616885366955359640013972, 17.83168540153277336650137255321, 18.31092880374162731984587372815, 19.95717212234859930560859199108, 20.69689804174122383703402417531, 21.50449922995641886975316429467, 22.64134914634724398814400802092, 23.26114950994563893162313786517, 24.04998805325050426759999448172, 25.12475178080954816575648896716, 26.63109991803076506574031704123

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