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Properties

Label 1-6005-6005.4263-r0-0-0
Degree $1$
Conductor $6005$
Sign $-0.0730 + 0.997i$
Analytic cond. $27.8871$
Root an. cond. $27.8871$
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  + (−0.309 − 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 6-s + (−0.707 + 0.707i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.852 − 0.522i)11-s + (−0.309 − 0.951i)12-s + (−0.233 − 0.972i)13-s + (0.891 + 0.453i)14-s + (0.309 − 0.951i)16-s + (−0.0784 + 0.996i)17-s + (−0.309 + 0.951i)18-s + (0.453 + 0.891i)19-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 6-s + (−0.707 + 0.707i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.852 − 0.522i)11-s + (−0.309 − 0.951i)12-s + (−0.233 − 0.972i)13-s + (0.891 + 0.453i)14-s + (0.309 − 0.951i)16-s + (−0.0784 + 0.996i)17-s + (−0.309 + 0.951i)18-s + (0.453 + 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0730 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0730 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6005\)    =    \(5 \cdot 1201\)
Sign: $-0.0730 + 0.997i$
Analytic conductor: \(27.8871\)
Root analytic conductor: \(27.8871\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6005} (4263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6005,\ (0:\ ),\ -0.0730 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4284911102 + 0.4610309076i\)
\(L(\frac12)\) \(\approx\) \(0.4284911102 + 0.4610309076i\)
\(L(1)\) \(\approx\) \(0.6594628425 + 0.003708057826i\)
\(L(1)\) \(\approx\) \(0.6594628425 + 0.003708057826i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1201 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.852 - 0.522i)T \)
13 \( 1 + (-0.233 - 0.972i)T \)
17 \( 1 + (-0.0784 + 0.996i)T \)
19 \( 1 + (0.453 + 0.891i)T \)
23 \( 1 + (-0.233 - 0.972i)T \)
29 \( 1 + (-0.760 + 0.649i)T \)
31 \( 1 + (-0.522 + 0.852i)T \)
37 \( 1 + (0.996 + 0.0784i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
43 \( 1 + (0.156 - 0.987i)T \)
47 \( 1 + (0.996 + 0.0784i)T \)
53 \( 1 + (0.522 - 0.852i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (0.987 + 0.156i)T \)
71 \( 1 + (-0.996 + 0.0784i)T \)
73 \( 1 + (0.852 - 0.522i)T \)
79 \( 1 + (-0.891 + 0.453i)T \)
83 \( 1 + (-0.382 - 0.923i)T \)
89 \( 1 + (-0.587 + 0.809i)T \)
97 \( 1 + (-0.923 - 0.382i)T \)
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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

−17.34440412616385556198124824005, −16.915202155182239524750069493039, −16.52498460925296977444739473788, −15.676211633633226811254216361652, −14.97563156923509600007734438015, −14.0209482022447883317308675310, −13.79571358738605190442004521849, −13.16258463706327359420101030956, −12.43174460923460160640297590643, −11.492526576647780407118322238496, −11.151647060691474523369749894583, −9.859364167285231440126884860323, −9.46178134188603626882993947385, −8.942609186490857215251012769995, −7.75342691821022885675654519800, −7.33579582009580595232522366559, −6.90554453500255646158890657833, −6.24662052041952008352832268548, −5.617410071788875329192981427768, −4.62948808382435352471380996099, −4.10563862102530793399534820272, −2.9951290581388589495850917850, −1.90943271935405866019109180271, −1.13749374476335895457659856439, −0.26431603975060816355167381464, 0.802386205472368709936551350618, 1.87171164889075712491726405360, 2.82141734929437680738356581371, 3.51195854557667897159067357424, 3.838234843423984924898801211093, 4.836786093339602337869527701406, 5.66287541832459783418787953512, 6.059389704687798142864673817134, 7.18733990096635103722742901093, 8.34068584269206763657842555258, 8.709475532718453969327183402022, 9.3474403318640596107758466102, 10.07780035426509783400561660005, 10.50627284976364102343866997493, 11.13955649256869564417050015960, 12.0311677008254543497307422556, 12.34706323532881782031950984532, 13.00377674467589722943436231984, 14.00163784243441618083358842190, 14.63616875851721689786286285158, 15.250544419727110919139027345021, 16.10773666569110423211563315368, 16.761077008145715321486256970930, 17.05933009105157812172277376420, 18.04219760424389276738182116866

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