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Properties

Label 1-605-605.399-r0-0-0
Degree $1$
Conductor $605$
Sign $0.995 + 0.0933i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (0.736 + 0.676i)2-s + (0.809 − 0.587i)3-s + (0.0855 + 0.996i)4-s + (0.993 + 0.113i)6-s + (0.466 − 0.884i)7-s + (−0.610 + 0.791i)8-s + (0.309 − 0.951i)9-s + (0.654 + 0.755i)12-s + (0.921 − 0.389i)13-s + (0.941 − 0.336i)14-s + (−0.985 + 0.170i)16-s + (0.254 − 0.967i)17-s + (0.870 − 0.491i)18-s + (−0.362 + 0.931i)19-s + (−0.142 − 0.989i)21-s + ⋯
L(s)  = 1  + (0.736 + 0.676i)2-s + (0.809 − 0.587i)3-s + (0.0855 + 0.996i)4-s + (0.993 + 0.113i)6-s + (0.466 − 0.884i)7-s + (−0.610 + 0.791i)8-s + (0.309 − 0.951i)9-s + (0.654 + 0.755i)12-s + (0.921 − 0.389i)13-s + (0.941 − 0.336i)14-s + (−0.985 + 0.170i)16-s + (0.254 − 0.967i)17-s + (0.870 − 0.491i)18-s + (−0.362 + 0.931i)19-s + (−0.142 − 0.989i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.995 + 0.0933i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.995 + 0.0933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.909888744 + 0.1360911034i\)
\(L(\frac12)\) \(\approx\) \(2.909888744 + 0.1360911034i\)
\(L(1)\) \(\approx\) \(2.051848346 + 0.2214027761i\)
\(L(1)\) \(\approx\) \(2.051848346 + 0.2214027761i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.736 + 0.676i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.466 - 0.884i)T \)
13 \( 1 + (0.921 - 0.389i)T \)
17 \( 1 + (0.254 - 0.967i)T \)
19 \( 1 + (-0.362 + 0.931i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (-0.998 - 0.0570i)T \)
31 \( 1 + (0.516 + 0.856i)T \)
37 \( 1 + (-0.974 + 0.226i)T \)
41 \( 1 + (0.198 + 0.980i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (0.870 + 0.491i)T \)
53 \( 1 + (0.985 + 0.170i)T \)
59 \( 1 + (0.198 - 0.980i)T \)
61 \( 1 + (-0.736 + 0.676i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (0.774 + 0.633i)T \)
73 \( 1 + (-0.696 + 0.717i)T \)
79 \( 1 + (-0.564 + 0.825i)T \)
83 \( 1 + (-0.897 + 0.441i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
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

−22.81586495557052424577674588509, −21.96982902717905786643707999358, −21.24162257323259758937763250024, −20.94363332211437064609208167994, −19.87031043358381139611043660448, −19.12445041912733727620939637471, −18.52902317017002125134811478763, −17.26433260569890779486078747968, −15.87576560326393871108253324749, −15.312528161074000786904859245688, −14.70055090614528997998219602669, −13.727639561443832386583055837065, −13.11866789529318428275624694552, −12.0214524830089600710729358934, −11.10965048488026668371773842778, −10.45139836038498918300989349068, −9.19709947412843084487548927982, −8.828790155886365617889194684929, −7.53428452539942662657766219560, −6.08225477068659513974619761401, −5.28173735541511475715691003287, −4.22416891097632380198340672586, −3.494711104218104001061301816318, −2.37888112104043493473409921373, −1.62123860958457506032095158965, 1.208905471825322831327709868536, 2.59027712938657402843777019306, 3.59713940870686031991828986613, 4.35128095409044890544412807301, 5.619840439187193079448924566737, 6.665756799051073268145276786159, 7.40961509009940855962367349778, 8.17506969310144514183268314835, 8.896407506262158831362359525301, 10.28241249203326245487183924195, 11.41886493789870671837488737200, 12.44043731422448461556173358768, 13.16661293744989396814303071380, 14.02965603035934961868490066719, 14.37924797140599365815475335255, 15.39818547591899362671468571022, 16.2791323584798222513846566771, 17.18856264538281685808835088262, 18.08252984387026343035077396833, 18.804629750090249787401162906107, 20.13697127131730300896362999007, 20.70276260830671377452192308336, 21.1777483201842583433792164636, 22.67602173848828192384290390496, 23.1414018544423271676114833860

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