Properties

Label 1-373-373.287-r0-0-0
Degree $1$
Conductor $373$
Sign $0.885 + 0.464i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (0.0506 − 0.998i)2-s + (−0.874 + 0.485i)3-s + (−0.994 − 0.101i)4-s + (0.758 + 0.651i)5-s + (0.440 + 0.897i)6-s + (0.820 + 0.571i)7-s + (−0.151 + 0.988i)8-s + (0.528 − 0.848i)9-s + (0.688 − 0.724i)10-s + (−0.151 − 0.988i)11-s + (0.918 − 0.394i)12-s + (0.151 + 0.988i)13-s + (0.612 − 0.790i)14-s + (−0.979 − 0.201i)15-s + (0.979 + 0.201i)16-s + (−0.994 − 0.101i)17-s + ⋯
L(s)  = 1  + (0.0506 − 0.998i)2-s + (−0.874 + 0.485i)3-s + (−0.994 − 0.101i)4-s + (0.758 + 0.651i)5-s + (0.440 + 0.897i)6-s + (0.820 + 0.571i)7-s + (−0.151 + 0.988i)8-s + (0.528 − 0.848i)9-s + (0.688 − 0.724i)10-s + (−0.151 − 0.988i)11-s + (0.918 − 0.394i)12-s + (0.151 + 0.988i)13-s + (0.612 − 0.790i)14-s + (−0.979 − 0.201i)15-s + (0.979 + 0.201i)16-s + (−0.994 − 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.885 + 0.464i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.885 + 0.464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9517929226 + 0.2345140985i\)
\(L(\frac12)\) \(\approx\) \(0.9517929226 + 0.2345140985i\)
\(L(1)\) \(\approx\) \(0.8868241144 - 0.06118758378i\)
\(L(1)\) \(\approx\) \(0.8868241144 - 0.06118758378i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.0506 - 0.998i)T \)
3 \( 1 + (-0.874 + 0.485i)T \)
5 \( 1 + (0.758 + 0.651i)T \)
7 \( 1 + (0.820 + 0.571i)T \)
11 \( 1 + (-0.151 - 0.988i)T \)
13 \( 1 + (0.151 + 0.988i)T \)
17 \( 1 + (-0.994 - 0.101i)T \)
19 \( 1 + (-0.979 + 0.201i)T \)
23 \( 1 + (0.874 + 0.485i)T \)
29 \( 1 + (-0.250 + 0.968i)T \)
31 \( 1 + (0.918 + 0.394i)T \)
37 \( 1 + (-0.954 - 0.299i)T \)
41 \( 1 + (-0.440 - 0.897i)T \)
43 \( 1 + (0.994 + 0.101i)T \)
47 \( 1 + (0.612 + 0.790i)T \)
53 \( 1 + (-0.528 - 0.848i)T \)
59 \( 1 + (-0.250 + 0.968i)T \)
61 \( 1 + (0.874 - 0.485i)T \)
67 \( 1 + (0.758 + 0.651i)T \)
71 \( 1 + (-0.994 + 0.101i)T \)
73 \( 1 + (0.528 + 0.848i)T \)
79 \( 1 + (-0.688 + 0.724i)T \)
83 \( 1 + (0.528 + 0.848i)T \)
89 \( 1 + T \)
97 \( 1 + (0.994 + 0.101i)T \)
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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

−24.66774013341117910724885225548, −23.72706349604584184593883367994, −23.042218403552342940615825797158, −22.25478656743239936016133131994, −21.20671638555861978360858175410, −20.29927476404884974128546301174, −18.891840091815859200737978620014, −17.799407060313347939915080022136, −17.38554248043440126811972178767, −16.99136387340243971909155246676, −15.7043839548050154570361331726, −14.89086559173660570667777917855, −13.56870721774493638986462485088, −13.11228538037773233358712367087, −12.24865185997776175943083029102, −10.78474778948207413813435158759, −9.99883611589519824684966468529, −8.6692453482197266579481648864, −7.7997949663581336535268757426, −6.80699173872741993381400192065, −5.95422732124938375125777285296, −4.85780124348115953449918901136, −4.480637468226574767119978855278, −2.04470238867673549255382104149, −0.715943377250268704996659367097, 1.41764407350882715674399729078, 2.51303390392174396071963569057, 3.8211491221997755973326514115, 4.938115494170481804100826238589, 5.73335026401042684205380911680, 6.7808007255958692800954235624, 8.67878392894114062501329733706, 9.26012943694010968604459098115, 10.580642263019961796277392444115, 10.99762524888722661802138382710, 11.712117677694389621741645667049, 12.83212434310816180991756721755, 13.90243700109236525372032778384, 14.658410262102856413504427649747, 15.784492709768946399307060367425, 17.18905550440448832638394706043, 17.610199588872507845038347014505, 18.63372852310045874342079901937, 19.11379360754637433355065139910, 20.8332828226950470585654761099, 21.28996494233062602888999483164, 21.851158259774076813683100647353, 22.56680969937792663117904740987, 23.637125454405826370508922636428, 24.356372990658440761731869446898

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