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Properties

Label 1-373-373.148-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.961 + 0.274i$
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  + (−0.378 − 0.925i)2-s + (0.217 − 0.975i)3-s + (−0.713 + 0.701i)4-s + (−0.315 + 0.948i)5-s + (−0.985 + 0.168i)6-s + (−0.0506 + 0.998i)7-s + (0.918 + 0.394i)8-s + (−0.905 − 0.425i)9-s + (0.997 − 0.0675i)10-s + (−0.801 − 0.598i)11-s + (0.528 + 0.848i)12-s + (0.918 − 0.394i)13-s + (0.943 − 0.331i)14-s + (0.857 + 0.514i)15-s + (0.0168 − 0.999i)16-s + (−0.250 − 0.968i)17-s + ⋯
L(s)  = 1  + (−0.378 − 0.925i)2-s + (0.217 − 0.975i)3-s + (−0.713 + 0.701i)4-s + (−0.315 + 0.948i)5-s + (−0.985 + 0.168i)6-s + (−0.0506 + 0.998i)7-s + (0.918 + 0.394i)8-s + (−0.905 − 0.425i)9-s + (0.997 − 0.0675i)10-s + (−0.801 − 0.598i)11-s + (0.528 + 0.848i)12-s + (0.918 − 0.394i)13-s + (0.943 − 0.331i)14-s + (0.857 + 0.514i)15-s + (0.0168 − 0.999i)16-s + (−0.250 − 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.961 + 0.274i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.961 + 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05985620552 - 0.4275794179i\)
\(L(\frac12)\) \(\approx\) \(-0.05985620552 - 0.4275794179i\)
\(L(1)\) \(\approx\) \(0.5094218396 - 0.3909308454i\)
\(L(1)\) \(\approx\) \(0.5094218396 - 0.3909308454i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.378 - 0.925i)T \)
3 \( 1 + (0.217 - 0.975i)T \)
5 \( 1 + (-0.315 + 0.948i)T \)
7 \( 1 + (-0.0506 + 0.998i)T \)
11 \( 1 + (-0.801 - 0.598i)T \)
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 \( 1 + (-0.931 + 0.363i)T \)
31 \( 1 + (0.528 - 0.848i)T \)
37 \( 1 + (0.283 - 0.959i)T \)
41 \( 1 + (0.347 - 0.937i)T \)
43 \( 1 + (-0.713 + 0.701i)T \)
47 \( 1 + (-0.184 - 0.982i)T \)
53 \( 1 + (-0.905 + 0.425i)T \)
59 \( 1 + (0.780 + 0.625i)T \)
61 \( 1 + (0.217 - 0.975i)T \)
67 \( 1 + (0.979 - 0.201i)T \)
71 \( 1 + (0.963 - 0.266i)T \)
73 \( 1 + (0.0843 - 0.996i)T \)
79 \( 1 + (0.997 - 0.0675i)T \)
83 \( 1 + (-0.905 + 0.425i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−25.43448880024198345948786519308, −24.04212691444727823842654539689, −23.50594280843605331032937908294, −22.81864523382720307517132073205, −21.49770575081118565370228619226, −20.60211987709845503356214960806, −19.93335782720695015888693304633, −18.9924359880199491919192325346, −17.59847239444260969770936742605, −16.940996254557371994731977493957, −16.17762813644626948784538851354, −15.58402220913726868820648549437, −14.65450154819058673738848198444, −13.61445254084481179576497665760, −12.867659987823928488145318562032, −11.17472786788518062511773901368, −10.24838767325445797484417834299, −9.55975009478513174433694768767, −8.295555140802421998879325726238, −8.052134756172861954694219476561, −6.51162032940520386177165313003, −5.391572276153495855045120215240, −4.35247028745151863028325461558, −3.87861316233521076027880633307, −1.58177376665361208686508449637, 0.28349862699900650987891562772, 2.13638771436060662735287933611, 2.70472714526342532626868873930, 3.701052555356169916723630860343, 5.49289293374926638442301284910, 6.57232532006124540215348473025, 7.83116809561009541255116717437, 8.414544668625405490667862507440, 9.45523138164787414142478628653, 10.86439149870762221450618033380, 11.353078165004108253661721932238, 12.33339070397124135897914837077, 13.21579458196636110215164020709, 13.965282157471736486779604119727, 15.1161843102068853131031435493, 16.18326713972997869419358428188, 17.68342783094456310672523454647, 18.4070805006138875020648499450, 18.65134931949150435670437107045, 19.52932572759972123365453034080, 20.49053925825215736156035346357, 21.442403391610238348368061840189, 22.39666795825943258038842115376, 23.07305346128758385305644153083, 24.057079207931026389110237032153

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