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Properties

Label 1-4235-4235.1952-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.966 + 0.256i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.226 + 0.974i)2-s + (−0.951 + 0.309i)3-s + (−0.897 − 0.441i)4-s + (−0.0855 − 0.996i)6-s + (0.633 − 0.774i)8-s + (0.809 − 0.587i)9-s + (0.989 + 0.142i)12-s + (0.884 − 0.466i)13-s + (0.610 + 0.791i)16-s + (−0.980 − 0.198i)17-s + (0.389 + 0.921i)18-s + (0.993 + 0.113i)19-s + (−0.281 + 0.959i)23-s + (−0.362 + 0.931i)24-s + (0.254 + 0.967i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (−0.226 + 0.974i)2-s + (−0.951 + 0.309i)3-s + (−0.897 − 0.441i)4-s + (−0.0855 − 0.996i)6-s + (0.633 − 0.774i)8-s + (0.809 − 0.587i)9-s + (0.989 + 0.142i)12-s + (0.884 − 0.466i)13-s + (0.610 + 0.791i)16-s + (−0.980 − 0.198i)17-s + (0.389 + 0.921i)18-s + (0.993 + 0.113i)19-s + (−0.281 + 0.959i)23-s + (−0.362 + 0.931i)24-s + (0.254 + 0.967i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.966 + 0.256i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1952, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.966 + 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07691542239 + 0.5904203449i\)
\(L(\frac12)\) \(\approx\) \(0.07691542239 + 0.5904203449i\)
\(L(1)\) \(\approx\) \(0.5363783635 + 0.3266082362i\)
\(L(1)\) \(\approx\) \(0.5363783635 + 0.3266082362i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.226 + 0.974i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (0.884 - 0.466i)T \)
17 \( 1 + (-0.980 - 0.198i)T \)
19 \( 1 + (0.993 + 0.113i)T \)
23 \( 1 + (-0.281 + 0.959i)T \)
29 \( 1 + (0.736 + 0.676i)T \)
31 \( 1 + (-0.696 - 0.717i)T \)
37 \( 1 + (0.170 - 0.985i)T \)
41 \( 1 + (-0.516 + 0.856i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 + (-0.389 + 0.921i)T \)
53 \( 1 + (-0.791 - 0.610i)T \)
59 \( 1 + (0.516 + 0.856i)T \)
61 \( 1 + (-0.974 + 0.226i)T \)
67 \( 1 + (0.755 - 0.654i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (-0.825 - 0.564i)T \)
79 \( 1 + (0.998 - 0.0570i)T \)
83 \( 1 + (-0.336 + 0.941i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.931 + 0.362i)T \)
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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

−18.2569840409174651083573518494, −17.54515468177802829333207166816, −16.93750914326450330463121030555, −16.16642778262782146938278937390, −15.637243714560346731158422572428, −14.417500672898714535196575561835, −13.66280128777573971393859472302, −13.177518087096450391321174947080, −12.480433414618927895688991695005, −11.666718477414946942924501045128, −11.42294392094500988883909154098, −10.5446021888919158761532987135, −10.12323746102127637410840720975, −9.16336970467521984015289572048, −8.49533409670470571874535078468, −7.73248729027343992470216438506, −6.76456152697747536354077307429, −6.19218493420916212375483746674, −5.15863177435896406918679058794, −4.59128732208248839159276950315, −3.835779827781378733461051218041, −2.89838087321713387491524909193, −1.896909142515195644413159086028, −1.30479850158694845570927252887, −0.27748947889145442349944334298, 0.864084669531028014480446936988, 1.669311165827020452390454779718, 3.27911401399134220796055383289, 3.99423744946235096893606262949, 4.81672287711318851571347742201, 5.46575178238003170740185404312, 6.04142303874244408218584546385, 6.74128931360753958060913989260, 7.4290859844897810000603288092, 8.18300934704654395412370319604, 9.12452248903846719808167954829, 9.58630349357175949114992315818, 10.43811483851483995309928380180, 11.043539687812586255793786290362, 11.75199806126663844942857275351, 12.700280006277874419223858829, 13.34251604621235148460003632613, 13.93938841229493701320505663595, 14.94697188017551460955749854913, 15.4722265866807778999372077763, 16.15326811292873776621026854668, 16.41197629812547596861197458792, 17.40492369615886904795984783954, 17.99116534524562304275122541107, 18.154488335275757504765903371071

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