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Properties

Label 1-373-373.85-r1-0-0
Degree $1$
Conductor $373$
Sign $0.981 - 0.190i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (0.266 − 0.963i)2-s + (−0.0843 − 0.996i)3-s + (−0.857 − 0.514i)4-s + (0.993 + 0.117i)5-s + (−0.982 − 0.184i)6-s + (0.994 + 0.101i)7-s + (−0.724 + 0.688i)8-s + (−0.985 + 0.168i)9-s + (0.378 − 0.925i)10-s + (−0.234 + 0.972i)11-s + (−0.440 + 0.897i)12-s + (−0.688 + 0.724i)13-s + (0.363 − 0.931i)14-s + (0.0337 − 0.999i)15-s + (0.470 + 0.882i)16-s + (0.874 − 0.485i)17-s + ⋯
L(s)  = 1  + (0.266 − 0.963i)2-s + (−0.0843 − 0.996i)3-s + (−0.857 − 0.514i)4-s + (0.993 + 0.117i)5-s + (−0.982 − 0.184i)6-s + (0.994 + 0.101i)7-s + (−0.724 + 0.688i)8-s + (−0.985 + 0.168i)9-s + (0.378 − 0.925i)10-s + (−0.234 + 0.972i)11-s + (−0.440 + 0.897i)12-s + (−0.688 + 0.724i)13-s + (0.363 − 0.931i)14-s + (0.0337 − 0.999i)15-s + (0.470 + 0.882i)16-s + (0.874 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.981 - 0.190i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.981 - 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.762154716 - 0.1690472716i\)
\(L(\frac12)\) \(\approx\) \(1.762154716 - 0.1690472716i\)
\(L(1)\) \(\approx\) \(1.052160220 - 0.5933571843i\)
\(L(1)\) \(\approx\) \(1.052160220 - 0.5933571843i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.266 - 0.963i)T \)
3 \( 1 + (-0.0843 - 0.996i)T \)
5 \( 1 + (0.993 + 0.117i)T \)
7 \( 1 + (0.994 + 0.101i)T \)
11 \( 1 + (-0.234 + 0.972i)T \)
13 \( 1 + (-0.688 + 0.724i)T \)
17 \( 1 + (0.874 - 0.485i)T \)
19 \( 1 + (-0.848 + 0.528i)T \)
23 \( 1 + (-0.571 + 0.820i)T \)
29 \( 1 + (0.217 + 0.975i)T \)
31 \( 1 + (0.440 + 0.897i)T \)
37 \( 1 + (-0.890 + 0.455i)T \)
41 \( 1 + (-0.758 - 0.651i)T \)
43 \( 1 + (-0.514 + 0.857i)T \)
47 \( 1 + (-0.625 + 0.780i)T \)
53 \( 1 + (-0.168 + 0.985i)T \)
59 \( 1 + (-0.736 + 0.676i)T \)
61 \( 1 + (0.996 - 0.0843i)T \)
67 \( 1 + (-0.394 - 0.918i)T \)
71 \( 1 + (-0.0168 - 0.999i)T \)
73 \( 1 + (0.638 - 0.769i)T \)
79 \( 1 + (0.925 + 0.378i)T \)
83 \( 1 + (-0.985 - 0.168i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.485 + 0.874i)T \)
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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

−24.49302502355186954190945310757, −23.66842783455245234902151208922, −22.58588107509039421773162981443, −21.726005881854215990614072840, −21.29681195140593581309637799426, −20.47767033363311224045740038476, −18.91659411043000672520201772906, −17.75682064694583448347216067266, −17.14076390740044810913791932569, −16.590046091959140952676405267665, −15.405410546560678620681897193924, −14.66258273432142660778325029282, −14.02427127700926100600754663228, −13.06055200565112656412873155025, −11.79271335358314137776855301630, −10.49007137332239492885322121723, −9.83916416795799920426592931407, −8.551858740283795605097466413314, −8.12171348765839116240730298593, −6.45850137603815821414603487500, −5.5177817701992873588238115278, −4.98404728265366840265077930436, −3.84917571263862567059598703268, −2.516571446562229950098348094678, −0.43847825905806058120677690065, 1.513532068405684921610064012233, 1.83845312915857383049058480457, 2.975605808580563862996473013847, 4.74483046841105434857230214777, 5.41290323140819098499974237739, 6.626857768924505657105259888640, 7.805318972158747230134420098132, 8.92920100247585589182494901224, 9.94817098572806794119184651675, 10.84437022997968978233244929877, 12.06714937196731535982794870159, 12.37636840053648488970187853334, 13.62763349280210316986314194171, 14.2141872678021138507689037660, 14.86939935322000440887403569123, 16.92125252547830479108356606412, 17.66859435298026572943380389992, 18.22218030216271172890540884875, 19.04109929994763236539066442567, 20.04231511685382775922442203238, 20.94985493479919254252450916318, 21.55463953735647627500821765070, 22.599479039034582472804053096789, 23.48293391752075096750558218156, 24.15815184895579947874951159508

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