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Properties

Label 1-6076-6076.4315-r1-0-0
Degree $1$
Conductor $6076$
Sign $-0.223 - 0.974i$
Analytic cond. $652.956$
Root an. cond. $652.956$
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  + (−0.623 + 0.781i)3-s + (−0.623 + 0.781i)5-s + (−0.222 − 0.974i)9-s + (−0.733 − 0.680i)11-s + (0.955 − 0.294i)13-s + (−0.222 − 0.974i)15-s + (−0.900 + 0.433i)17-s + (−0.5 − 0.866i)19-s + (0.826 − 0.563i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + (0.900 − 0.433i)29-s + (0.988 − 0.149i)33-s + (−0.826 − 0.563i)37-s + (−0.365 + 0.930i)39-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)3-s + (−0.623 + 0.781i)5-s + (−0.222 − 0.974i)9-s + (−0.733 − 0.680i)11-s + (0.955 − 0.294i)13-s + (−0.222 − 0.974i)15-s + (−0.900 + 0.433i)17-s + (−0.5 − 0.866i)19-s + (0.826 − 0.563i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + (0.900 − 0.433i)29-s + (0.988 − 0.149i)33-s + (−0.826 − 0.563i)37-s + (−0.365 + 0.930i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6076 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6076 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6076\)    =    \(2^{2} \cdot 7^{2} \cdot 31\)
Sign: $-0.223 - 0.974i$
Analytic conductor: \(652.956\)
Root analytic conductor: \(652.956\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6076} (4315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6076,\ (1:\ ),\ -0.223 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4875061861 - 0.6118296857i\)
\(L(\frac12)\) \(\approx\) \(0.4875061861 - 0.6118296857i\)
\(L(1)\) \(\approx\) \(0.7058242143 + 0.09671821149i\)
\(L(1)\) \(\approx\) \(0.7058242143 + 0.09671821149i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
31 \( 1 \)
good3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (0.988 + 0.149i)T \)
43 \( 1 + (-0.988 + 0.149i)T \)
47 \( 1 + (-0.733 - 0.680i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.0747 - 0.997i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.733 - 0.680i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 - T \)
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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

−17.69323441004060780407487482338, −17.17192984883470477583077510316, −16.33442227852994484737406931185, −15.91350830005189090705687544468, −15.33622654409453962748459403006, −14.33718935603961027732686983837, −13.547961605242774667239903787640, −12.91316532833125888727856658624, −12.63634648683350210358239043152, −11.75618435365615088528864359106, −11.300779938989037359515707450039, −10.63275156132363150555969070629, −9.80122545674006830712719844936, −8.800912258018706511307083358383, −8.324994871404316171224434178879, −7.673285673390314690293329330304, −6.90999804549100642580185928376, −6.38006539240921479220672511535, −5.3746856243729320779233507821, −4.924012215200606218600384649, −4.20349382838913009921505539974, −3.26730312742397867237588302055, −2.224621301629506573481621859164, −1.48561455512397504785540624093, −0.79874915364131470325846356034, 0.20029503511161334218479366864, 0.6994074223159143384016712918, 2.15997229954988058722876277762, 3.098376630718827774660383303411, 3.501871244015183531140727995669, 4.44620214950695768007927175699, 4.92200111085667037798661760813, 5.96775631302686084547610773773, 6.42743375911896191558870321157, 7.05890868755683822455991011082, 8.178610173737500757611000750837, 8.58395495118146017388503368344, 9.38491192186635873438514919447, 10.45622516736148423042764229450, 10.74525317881774182794835792243, 11.15847767211734946731883545464, 11.836175482555071647578881630592, 12.74845607528991759474085219619, 13.36509131093109660037803791074, 14.19087990918630600826707346085, 15.0005659195750204918136429192, 15.51121296913912832963928414528, 15.86424559310861662235446780171, 16.555783120244162913424382908052, 17.3977232968269014106481666397

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