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Properties

Label 1-273-273.173-r0-0-0
Degree $1$
Conductor $273$
Sign $0.113 + 0.993i$
Analytic cond. $1.26780$
Root an. cond. $1.26780$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + 8-s − 10-s + 11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 19-s + (0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + 8-s − 10-s + 11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 19-s + (0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.113 + 0.993i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ 0.113 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7862780271 + 0.7014407798i\)
\(L(\frac12)\) \(\approx\) \(0.7862780271 + 0.7014407798i\)
\(L(1)\) \(\approx\) \(0.8223794602 + 0.4413038549i\)
\(L(1)\) \(\approx\) \(0.8223794602 + 0.4413038549i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
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

−25.52705356987418742139319281936, −24.80009575731503859584061432066, −23.715942874507777787999628982464, −22.3863068855410913070335796251, −21.75368059839593282158103682576, −20.808179497849611274781971029602, −20.02184594320122868028717226105, −19.31441902105001137818274740260, −18.13920649401333716754819953434, −17.199488820811048782268909668003, −16.75911836588648539845563455686, −15.42013886027548872052303279171, −13.89891157433594952432101093691, −13.230733083445923232751316352002, −12.17519991880555906637590728751, −11.453842915890571820019214961664, −10.17441300912329104190440327766, −9.30847266352030864406424930911, −8.62837017457625177260093614210, −7.42684869239547588085328175128, −5.92898812219355534319685760728, −4.6115299849172322941682848238, −3.607098443385230170911315747939, −2.07713191214524113224743176461, −1.05698514096191191825271671473, 1.3367236144619198973623309464, 2.92967537031707226145455900978, 4.52309706130405256505194939550, 5.7436342557686966339536914636, 6.73613622330892118433743525320, 7.35304409309567663850772939538, 8.82566618121164030409503021102, 9.55401150280083880127502671242, 10.58196793537489480320152121284, 11.54961841742825802669247348514, 13.143765829157688528861824921517, 14.289370580900997626182794967638, 14.57373854335384598648780113411, 15.87671189674795892793194827496, 16.66693282329150833296537176294, 17.85429782927654020834980417495, 18.20064665628938882888657097259, 19.33850956941159961039859064130, 20.1750473659336190021083305832, 21.66898569379193021361609203602, 22.52634938864567973982524424560, 23.1178608059171368038884495645, 24.49462510196710209859847567893, 25.006208017576185787115189133343, 25.88382119170862102050794223768

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