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Properties

Label 1-147-147.131-r0-0-0
Degree $1$
Conductor $147$
Sign $-0.274 - 0.961i$
Analytic cond. $0.682665$
Root an. cond. $0.682665$
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  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.365 − 0.930i)5-s + (−0.623 − 0.781i)8-s + (−0.365 − 0.930i)10-s + (−0.955 − 0.294i)11-s + (0.222 + 0.974i)13-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (0.5 + 0.866i)19-s + (−0.900 − 0.433i)20-s + (−0.900 + 0.433i)22-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)25-s + (0.826 + 0.563i)26-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.365 − 0.930i)5-s + (−0.623 − 0.781i)8-s + (−0.365 − 0.930i)10-s + (−0.955 − 0.294i)11-s + (0.222 + 0.974i)13-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (0.5 + 0.866i)19-s + (−0.900 − 0.433i)20-s + (−0.900 + 0.433i)22-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)25-s + (0.826 + 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.274 - 0.961i$
Analytic conductor: \(0.682665\)
Root analytic conductor: \(0.682665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (0:\ ),\ -0.274 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9539142248 - 1.264012437i\)
\(L(\frac12)\) \(\approx\) \(0.9539142248 - 1.264012437i\)
\(L(1)\) \(\approx\) \(1.212888904 - 0.8509685403i\)
\(L(1)\) \(\approx\) \(1.212888904 - 0.8509685403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.733 - 0.680i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (-0.955 - 0.294i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.826 - 0.563i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.0747 + 0.997i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.733 + 0.680i)T \)
53 \( 1 + (-0.0747 + 0.997i)T \)
59 \( 1 + (0.365 + 0.930i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
97 \( 1 - T \)
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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

−28.57767260936215006824324114203, −27.18940635299463979817406411292, −26.091524149720708480714315110133, −25.66483524563453595432667659435, −24.53530546535108807778448598345, −23.35156665176225905113561986728, −22.765291594174340932674520508053, −21.6701534983789067801384761991, −20.96211485977345153328263820552, −19.55551855642417762294498697272, −17.98051657913535234707953395808, −17.65151295964547479004333608578, −16.02464702710992102495766591752, −15.311580375489218710652701923074, −14.28108036411424953646053068836, −13.38619827081863343786107722306, −12.35090115546267157771851416766, −10.99496778309166459371441058489, −9.9067092785613318163563320696, −8.16530689105506061559123009415, −7.304968169091681919483566022793, −6.07578195683502350319517245745, −5.16307307169443806077132359587, −3.52260367770827013390424600307, −2.51275274685182878601161809936, 1.24084796739326144460391683846, 2.65677420888612004036614857579, 4.18759882202652632607594221414, 5.23331472684184035312510893666, 6.22681221078013877293609106489, 8.04800657450612332349521591461, 9.42701644567610429250984465240, 10.30012468370741585300064331682, 11.70019270346270988875179422632, 12.48335805948632566675609246801, 13.58973028306174404853378185084, 14.263396000286986790970550108827, 15.82495576616540009754151875467, 16.55066386730067656096323393224, 18.18023255088620415499198459193, 19.015494041198674193096468059, 20.34406167219859810822008714865, 20.92734805944646987375453649744, 21.72522791640385525092673122149, 22.99162241593123749476022305323, 23.88549122537131974852105273034, 24.6084028063574095756325187074, 25.79966882626388875045441123166, 27.193643321647057941900815641072, 28.25453190751962218511522096393

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