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Properties

Label 1-276-276.131-r0-0-0
Degree $1$
Conductor $276$
Sign $0.991 + 0.130i$
Analytic cond. $1.28173$
Root an. cond. $1.28173$
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  + (0.654 − 0.755i)5-s + (−0.841 + 0.540i)7-s + (−0.142 + 0.989i)11-s + (0.841 + 0.540i)13-s + (0.959 + 0.281i)17-s + (0.959 − 0.281i)19-s + (−0.142 − 0.989i)25-s + (0.959 + 0.281i)29-s + (−0.415 − 0.909i)31-s + (−0.142 + 0.989i)35-s + (−0.654 − 0.755i)37-s + (0.654 − 0.755i)41-s + (−0.415 + 0.909i)43-s + 47-s + (0.415 − 0.909i)49-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)5-s + (−0.841 + 0.540i)7-s + (−0.142 + 0.989i)11-s + (0.841 + 0.540i)13-s + (0.959 + 0.281i)17-s + (0.959 − 0.281i)19-s + (−0.142 − 0.989i)25-s + (0.959 + 0.281i)29-s + (−0.415 − 0.909i)31-s + (−0.142 + 0.989i)35-s + (−0.654 − 0.755i)37-s + (0.654 − 0.755i)41-s + (−0.415 + 0.909i)43-s + 47-s + (0.415 − 0.909i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(1.28173\)
Root analytic conductor: \(1.28173\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 276,\ (0:\ ),\ 0.991 + 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.328737077 + 0.08717245512i\)
\(L(\frac12)\) \(\approx\) \(1.328737077 + 0.08717245512i\)
\(L(1)\) \(\approx\) \(1.152365792 + 0.01560872508i\)
\(L(1)\) \(\approx\) \(1.152365792 + 0.01560872508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (0.841 + 0.540i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + (-0.654 - 0.755i)T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.959 + 0.281i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−25.58724559095099592066663794412, −25.04833235447191832938476716945, −23.65288002325750973719743454028, −22.892277178672799521198894889647, −22.10339090714172572194303470757, −21.17314522465902481311482007871, −20.239076335465846455793563260584, −19.05913885894626873556529640477, −18.457229561789986901703493772, −17.42196053498092314663314395016, −16.35476653330774523980272324609, −15.64678500487244620097156361214, −14.14921087916073851411758397483, −13.75986606073473831336046137360, −12.69911651745718631580747389440, −11.360648362188466890035986722419, −10.39243788207569687888803359773, −9.74923478759935330812352663394, −8.42782345389639144352583502834, −7.21505721771558235399136476072, −6.23004391863059678232520364337, −5.411914693966189442083085605, −3.5094230113028027019234361292, −2.989320564097240540355245808052, −1.13311554044004588705649740250, 1.32738014525038000705084377054, 2.6183003441845657733636311583, 4.03025622486615248821521169384, 5.32287243993783016458586259726, 6.13145037292049337749470511009, 7.3576985081853750118274619511, 8.73708879451138363003331927850, 9.48142893926699530161859370947, 10.294960532841422506148380395519, 11.84599847850775344095912340800, 12.61177733066319422430322919968, 13.43002644136421695422951355726, 14.46592026102207193023099114761, 15.77539126812542272066548553983, 16.33398517498314943129994883859, 17.45200732363273905295706600892, 18.305598196079883553430635720077, 19.30369383455795420882442721560, 20.36820773322217968956098074178, 21.08677120105013818544640703160, 22.022522591837023829577107002228, 22.97733634033433019766230527652, 23.88623117053579634674233444751, 24.99504499395009936183014864630, 25.60002124381452224636208487078

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