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Properties

Label 1-373-373.13-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: $\chi_{373} (13, \cdot )$
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

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

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