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Properties

Label 1-373-373.147-r0-0-0
Degree $1$
Conductor $373$
Sign $0.457 - 0.889i$
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  + (−0.283 − 0.959i)2-s + (−0.713 − 0.701i)3-s + (−0.839 + 0.543i)4-s + (0.985 + 0.168i)5-s + (−0.470 + 0.882i)6-s + (−0.954 + 0.299i)7-s + (0.758 + 0.651i)8-s + (0.0168 + 0.999i)9-s + (−0.117 − 0.993i)10-s + (−0.943 − 0.331i)11-s + (0.979 + 0.201i)12-s + (−0.758 + 0.651i)13-s + (0.557 + 0.830i)14-s + (−0.585 − 0.810i)15-s + (0.409 − 0.912i)16-s + (−0.0506 − 0.998i)17-s + ⋯
L(s)  = 1  + (−0.283 − 0.959i)2-s + (−0.713 − 0.701i)3-s + (−0.839 + 0.543i)4-s + (0.985 + 0.168i)5-s + (−0.470 + 0.882i)6-s + (−0.954 + 0.299i)7-s + (0.758 + 0.651i)8-s + (0.0168 + 0.999i)9-s + (−0.117 − 0.993i)10-s + (−0.943 − 0.331i)11-s + (0.979 + 0.201i)12-s + (−0.758 + 0.651i)13-s + (0.557 + 0.830i)14-s + (−0.585 − 0.810i)15-s + (0.409 − 0.912i)16-s + (−0.0506 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.457 - 0.889i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.457 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6512221747 - 0.3974962691i\)
\(L(\frac12)\) \(\approx\) \(0.6512221747 - 0.3974962691i\)
\(L(1)\) \(\approx\) \(0.6331816758 - 0.3383623815i\)
\(L(1)\) \(\approx\) \(0.6331816758 - 0.3383623815i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.283 - 0.959i)T \)
3 \( 1 + (-0.713 - 0.701i)T \)
5 \( 1 + (0.985 + 0.168i)T \)
7 \( 1 + (-0.954 + 0.299i)T \)
11 \( 1 + (-0.943 - 0.331i)T \)
13 \( 1 + (-0.758 + 0.651i)T \)
17 \( 1 + (-0.0506 - 0.998i)T \)
19 \( 1 + (0.994 + 0.101i)T \)
23 \( 1 + (0.250 + 0.968i)T \)
29 \( 1 + (0.990 + 0.134i)T \)
31 \( 1 + (0.979 - 0.201i)T \)
37 \( 1 + (0.780 - 0.625i)T \)
41 \( 1 + (0.528 + 0.848i)T \)
43 \( 1 + (0.839 - 0.543i)T \)
47 \( 1 + (-0.997 - 0.0675i)T \)
53 \( 1 + (-0.0168 + 0.999i)T \)
59 \( 1 + (-0.378 - 0.925i)T \)
61 \( 1 + (0.713 + 0.701i)T \)
67 \( 1 + (-0.347 - 0.937i)T \)
71 \( 1 + (0.890 - 0.455i)T \)
73 \( 1 + (0.857 + 0.514i)T \)
79 \( 1 + (0.117 + 0.993i)T \)
83 \( 1 + (0.0168 - 0.999i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.0506 + 0.998i)T \)
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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

−24.79760891978869391955958351576, −23.94440520919461946041657532758, −22.87856573592567499593997394426, −22.43549020090120724549284642189, −21.53457609852821600361016667481, −20.49273619480631208045199774038, −19.352820010976916459599223970776, −18.09093326300841644047965410284, −17.56388013717879647984940934478, −16.76701105428682123170214217703, −16.02647909311542248972406000240, −15.25264503410674732977535053607, −14.24815988177040997079074550139, −13.10117709389063988024256227608, −12.491628476926833318670145643945, −10.58412650017207300728814666963, −10.05998958754993157394235539708, −9.49219656128879632909178765202, −8.23274941456508212957189049080, −6.89228599421696628021893127110, −6.10983075276905824821971804766, −5.28292174675619931918480242325, −4.45487331477582757122025845941, −2.87503462468817398747920683837, −0.78034388478401964148062862836, 0.92349488439374706888325448375, 2.34601906160676868020387991461, 2.94207719358883565138432238843, 4.860222936277807806131276427814, 5.65294798421463628673082029814, 6.83637782746761071486886794832, 7.84692228935017019156604753139, 9.35793044720918239327637011017, 9.854927031614802355500252411541, 10.911538107344407179349366442196, 11.8511307555361326545436570318, 12.6610585448137012585440001938, 13.51043039751054487381420942096, 14.004162021597188552899086696, 15.95296353542564146338633292533, 16.75615259510614751954146209647, 17.73840489384481031937262459212, 18.32615415473157727392130677909, 19.052865123522811219086656316172, 19.876253352227684018389940149436, 21.18579909304982915268477856192, 21.78140695425261970068054564553, 22.586524257702833305599687928539, 23.25778053293691420792121405670, 24.53019660114397634775312902500

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