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Properties

Label 1-4235-4235.1004-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.334 + 0.942i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.953 + 0.299i)2-s + (0.913 + 0.406i)3-s + (0.820 − 0.572i)4-s + (−0.993 − 0.113i)6-s + (−0.610 + 0.791i)8-s + (0.669 + 0.743i)9-s + (0.981 − 0.189i)12-s + (−0.921 + 0.389i)13-s + (0.345 − 0.938i)16-s + (0.964 − 0.263i)17-s + (−0.861 − 0.508i)18-s + (0.625 + 0.780i)19-s + (0.786 + 0.618i)23-s + (−0.879 + 0.475i)24-s + (0.761 − 0.647i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.953 + 0.299i)2-s + (0.913 + 0.406i)3-s + (0.820 − 0.572i)4-s + (−0.993 − 0.113i)6-s + (−0.610 + 0.791i)8-s + (0.669 + 0.743i)9-s + (0.981 − 0.189i)12-s + (−0.921 + 0.389i)13-s + (0.345 − 0.938i)16-s + (0.964 − 0.263i)17-s + (−0.861 − 0.508i)18-s + (0.625 + 0.780i)19-s + (0.786 + 0.618i)23-s + (−0.879 + 0.475i)24-s + (0.761 − 0.647i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.334 + 0.942i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.334 + 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.968434702 + 1.389499556i\)
\(L(\frac12)\) \(\approx\) \(1.968434702 + 1.389499556i\)
\(L(1)\) \(\approx\) \(1.002569719 + 0.3394912501i\)
\(L(1)\) \(\approx\) \(1.002569719 + 0.3394912501i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.953 + 0.299i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.921 + 0.389i)T \)
17 \( 1 + (0.964 - 0.263i)T \)
19 \( 1 + (0.625 + 0.780i)T \)
23 \( 1 + (0.786 + 0.618i)T \)
29 \( 1 + (-0.998 - 0.0570i)T \)
31 \( 1 + (-0.483 + 0.875i)T \)
37 \( 1 + (0.290 - 0.956i)T \)
41 \( 1 + (-0.198 - 0.980i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (0.861 - 0.508i)T \)
53 \( 1 + (-0.345 - 0.938i)T \)
59 \( 1 + (0.948 - 0.318i)T \)
61 \( 1 + (0.217 + 0.976i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (0.774 + 0.633i)T \)
73 \( 1 + (-0.969 - 0.244i)T \)
79 \( 1 + (-0.432 - 0.901i)T \)
83 \( 1 + (0.897 - 0.441i)T \)
89 \( 1 + (0.888 - 0.458i)T \)
97 \( 1 + (-0.0285 + 0.999i)T \)
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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

−18.338466281115405670710746802753, −17.409727900094591477314712059998, −16.95543152751003216916534736484, −16.13414140722845244870191639285, −15.24523645889738558510418574056, −14.87536975049241228416214362206, −14.07084897161385943012610368028, −13.03180979675342874548188133821, −12.70146952147963587474377702794, −11.91176436545111135225260476199, −11.166434512776392184824224989517, −10.329128447787526301432488965975, −9.50226098747581468650008402796, −9.317411578554623961010271590952, −8.30194543540162292197023000827, −7.732490564662443134127509844646, −7.25449439049964454330214562483, −6.49820572000158474949985223049, −5.51455107158528468578546537958, −4.39389715399592080715166646580, −3.42606603245649908737896472445, −2.80314022546317706850874451021, −2.21429862720632294878682412229, −1.18910968921310749513204229988, −0.59185312224251783477665976836, 0.69399869971114369422519109760, 1.69084450817858149504779808388, 2.3273757706573202500087314105, 3.233738594005068402079657595458, 3.92030156232903871529050444885, 5.23250746353403200145619092809, 5.50214698923357473752971614165, 6.854245265500278273969806282068, 7.49121007993307467513518299815, 7.780981688687881593054695883014, 8.91774255141902585032168844488, 9.1908584439051565394750980366, 9.97732387611196470316771311498, 10.42694449883996475794666980051, 11.336378315538054804101674839, 12.093074693751700678649604872438, 12.88034751667362999430195491515, 13.949043070831892513870732849852, 14.50897872723277618963656877679, 14.869053808066509297335634177208, 15.84065227074506339860910065421, 16.22997769858205117058285065338, 16.94147505352822700602857979089, 17.60057180431419252838817566431, 18.545227670110858714548548975758

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