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Properties

Label 1-373-373.361-r0-0-0
Degree $1$
Conductor $373$
Sign $0.528 + 0.849i$
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  + (0.612 + 0.790i)2-s + (−0.954 − 0.299i)3-s + (−0.250 + 0.968i)4-s + (−0.979 − 0.201i)5-s + (−0.347 − 0.937i)6-s + (−0.0506 − 0.998i)7-s + (−0.918 + 0.394i)8-s + (0.820 + 0.571i)9-s + (−0.440 − 0.897i)10-s + (−0.918 − 0.394i)11-s + (0.528 − 0.848i)12-s + (0.918 + 0.394i)13-s + (0.758 − 0.651i)14-s + (0.874 + 0.485i)15-s + (−0.874 − 0.485i)16-s + (−0.250 + 0.968i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.790i)2-s + (−0.954 − 0.299i)3-s + (−0.250 + 0.968i)4-s + (−0.979 − 0.201i)5-s + (−0.347 − 0.937i)6-s + (−0.0506 − 0.998i)7-s + (−0.918 + 0.394i)8-s + (0.820 + 0.571i)9-s + (−0.440 − 0.897i)10-s + (−0.918 − 0.394i)11-s + (0.528 − 0.848i)12-s + (0.918 + 0.394i)13-s + (0.758 − 0.651i)14-s + (0.874 + 0.485i)15-s + (−0.874 − 0.485i)16-s + (−0.250 + 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.528 + 0.849i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.528 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8430503884 + 0.4684239780i\)
\(L(\frac12)\) \(\approx\) \(0.8430503884 + 0.4684239780i\)
\(L(1)\) \(\approx\) \(0.8459338297 + 0.2959251574i\)
\(L(1)\) \(\approx\) \(0.8459338297 + 0.2959251574i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.612 + 0.790i)T \)
3 \( 1 + (-0.954 - 0.299i)T \)
5 \( 1 + (-0.979 - 0.201i)T \)
7 \( 1 + (-0.0506 - 0.998i)T \)
11 \( 1 + (-0.918 - 0.394i)T \)
13 \( 1 + (0.918 + 0.394i)T \)
17 \( 1 + (-0.250 + 0.968i)T \)
19 \( 1 + (0.874 - 0.485i)T \)
23 \( 1 + (0.954 - 0.299i)T \)
29 \( 1 + (0.151 + 0.988i)T \)
31 \( 1 + (0.528 + 0.848i)T \)
37 \( 1 + (0.688 - 0.724i)T \)
41 \( 1 + (0.347 + 0.937i)T \)
43 \( 1 + (0.250 - 0.968i)T \)
47 \( 1 + (0.758 + 0.651i)T \)
53 \( 1 + (-0.820 + 0.571i)T \)
59 \( 1 + (0.151 + 0.988i)T \)
61 \( 1 + (0.954 + 0.299i)T \)
67 \( 1 + (-0.979 - 0.201i)T \)
71 \( 1 + (-0.250 - 0.968i)T \)
73 \( 1 + (0.820 - 0.571i)T \)
79 \( 1 + (0.440 + 0.897i)T \)
83 \( 1 + (0.820 - 0.571i)T \)
89 \( 1 + T \)
97 \( 1 + (0.250 - 0.968i)T \)
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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

−24.1098598128030350102435118269, −23.17151043068616620844501358021, −22.78527044634875798848388378698, −22.05446288060600909304020498219, −20.88304193222675084499785558305, −20.516014440108190919356972899733, −18.98311474237457021245061475439, −18.546418015514451949084468035530, −17.74301827711324628255178498219, −15.98346826447436956542645226400, −15.64768176267480472290208374723, −14.882946016388898015018824965338, −13.3718128366032620059160043211, −12.55336114198034611924527559331, −11.63097601838623691080805076947, −11.27952009492520583364694264750, −10.18957615288084613371257812188, −9.24979231749213048134515045588, −7.833306603207703304025162271864, −6.4729731469366723399223217117, −5.46204554499623448865443210441, −4.7486809297423836520385651082, −3.5899776131580067490521653454, −2.56512172649110956039339013581, −0.78637674227435147831721047876, 0.92732552028908384552752973407, 3.21732608107502594957027424965, 4.26401327128057484386726882413, 5.02606058061684623766348042455, 6.188582523593227191796987201181, 7.12059178494842137683789038726, 7.77969989434363638339372816623, 8.82584046052596107750276769994, 10.69597853582673691285922794625, 11.19702340086974214735958815658, 12.37192405187014252730928676078, 13.09241186414733434682925306259, 13.85312958448122026792608673541, 15.1995129745089192226471868043, 16.14445048184954637333719618780, 16.42439614850966713262068696966, 17.50495592921225515909532180515, 18.333796504470792802259877333669, 19.39162086289375492172489435696, 20.600139791632017695827074593074, 21.47539012114766039017472499875, 22.52882478883882353116893648267, 23.317647709127225712581354695388, 23.731034965041594304572348141858, 24.23261868740860701840232074542

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