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Properties

Label 1-373-373.31-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: $\chi_{373} (31, \cdot )$
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

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

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