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Properties

Label 1-260-260.203-r1-0-0
Degree $1$
Conductor $260$
Sign $0.749 + 0.661i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(27.9408\)
Root analytic conductor: \(27.9408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (1:\ ),\ 0.749 + 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.531269330 + 0.5791910662i\)
\(L(\frac12)\) \(\approx\) \(1.531269330 + 0.5791910662i\)
\(L(1)\) \(\approx\) \(1.100013832 - 0.06848438428i\)
\(L(1)\) \(\approx\) \(1.100013832 - 0.06848438428i\)

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

−25.71029394874060362278721436067, −24.56287050508828012810801574824, −23.82553499755854667466557830876, −22.57758062388034668030820384918, −21.87148072165303221974442480728, −20.93474367388578343332252911870, −20.38645955132475285334021775807, −19.16445608678240506795977796913, −18.03623802960159572778775369296, −17.09889215197293621563151500781, −16.19646518687537598634424739343, −15.34148240801902485832963668027, −14.33030028095766132829822520307, −13.65949828872824639737074019796, −11.999950181950931539815185343607, −11.12807546322275273804934984286, −10.461127587090354492482843376416, −9.0550832755305850022499132907, −8.494502880575857814134002653453, −7.095329925411799514798548334213, −5.56452197319148322527485934678, −4.84088592193211925545893513168, −3.6582462784892471520852107554, −2.42628875686202208128203151567, −0.52613359841814474803149218489, 1.38458254153020742837683136599, 2.12809383581174961224316413218, 3.82346322631681364595739953600, 5.20950964770687709269784153104, 6.24555213591181888817993742124, 7.54672457164112666608106861351, 8.03237885382736404263470532171, 9.33721284235581223908088803519, 10.69399866622463628439822219813, 11.707209423099203064884647821381, 12.508845600997797267873568180617, 13.47429533610340444417105600670, 14.54031353009516741983625799848, 15.18455465276445182310407331726, 16.85152184843938587399304975386, 17.52625840492510481638412047217, 18.31947863672759484187227144458, 19.23058000310211323983564272828, 20.249267015804107636597031809740, 21.0016442956256747268902586197, 22.26681335469466282197931031303, 23.28483732561529762222143676266, 23.924356521253365573751054717850, 24.8221718637325501535007738182, 25.57093047059069494460845897380

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