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Properties

Label 1-1375-1375.739-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.950 + 0.312i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (−0.637 + 0.770i)2-s + (−0.728 − 0.684i)3-s + (−0.187 − 0.982i)4-s + (0.992 − 0.125i)6-s + (0.309 − 0.951i)7-s + (0.876 + 0.481i)8-s + (0.0627 + 0.998i)9-s + (−0.535 + 0.844i)12-s + (0.0627 + 0.998i)13-s + (0.535 + 0.844i)14-s + (−0.929 + 0.368i)16-s + (0.728 − 0.684i)17-s + (−0.809 − 0.587i)18-s + (0.992 − 0.125i)19-s + (−0.876 + 0.481i)21-s + ⋯
L(s)  = 1  + (−0.637 + 0.770i)2-s + (−0.728 − 0.684i)3-s + (−0.187 − 0.982i)4-s + (0.992 − 0.125i)6-s + (0.309 − 0.951i)7-s + (0.876 + 0.481i)8-s + (0.0627 + 0.998i)9-s + (−0.535 + 0.844i)12-s + (0.0627 + 0.998i)13-s + (0.535 + 0.844i)14-s + (−0.929 + 0.368i)16-s + (0.728 − 0.684i)17-s + (−0.809 − 0.587i)18-s + (0.992 − 0.125i)19-s + (−0.876 + 0.481i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.950 + 0.312i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.950 + 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.368827461 + 0.2190094332i\)
\(L(\frac12)\) \(\approx\) \(1.368827461 + 0.2190094332i\)
\(L(1)\) \(\approx\) \(0.7477156904 + 0.03955167439i\)
\(L(1)\) \(\approx\) \(0.7477156904 + 0.03955167439i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.637 + 0.770i)T \)
3 \( 1 + (-0.728 - 0.684i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (0.0627 + 0.998i)T \)
17 \( 1 + (0.728 - 0.684i)T \)
19 \( 1 + (0.992 - 0.125i)T \)
23 \( 1 + (0.929 + 0.368i)T \)
29 \( 1 + (0.425 - 0.904i)T \)
31 \( 1 + (0.876 + 0.481i)T \)
37 \( 1 + (-0.0627 - 0.998i)T \)
41 \( 1 + (-0.0627 - 0.998i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (0.992 + 0.125i)T \)
53 \( 1 + (0.992 + 0.125i)T \)
59 \( 1 + (0.968 + 0.248i)T \)
61 \( 1 + (-0.968 + 0.248i)T \)
67 \( 1 + (-0.728 + 0.684i)T \)
71 \( 1 + (0.728 + 0.684i)T \)
73 \( 1 + (-0.637 + 0.770i)T \)
79 \( 1 + (-0.876 + 0.481i)T \)
83 \( 1 + (0.876 + 0.481i)T \)
89 \( 1 + (0.535 + 0.844i)T \)
97 \( 1 + (0.187 + 0.982i)T \)
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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

−20.716968928538754617891606179983, −20.00786728348709000480824630973, −18.91447000009157025657213600152, −18.3441356609213780027835582198, −17.66177681292609885643488601598, −16.995031264756483263295412324260, −16.23452457767493952678210689563, −15.40882021516047974935530794779, −14.780506797528296315972311486200, −13.491594175269267302973529470182, −12.45739179813186693565726013302, −12.07233398640585467974910314220, −11.28581252292070466983100457778, −10.4726819371463829296092838861, −9.95085487229182707710472365467, −9.02343522757468630197892502367, −8.37384052916668783421736153988, −7.42536421834910041979452160396, −6.217900899794060542314254605302, −5.32972845872042868986314107225, −4.61997529677699802525751998293, −3.39069426768107550657453928211, −2.857088189931734255691454148594, −1.445902413828714420763948985196, −0.58345252128014220114103382058, 0.82053238301066063218673125589, 1.163145868318836908868951578056, 2.45377260033174950530928318366, 4.117045220330964370753791357098, 4.99724607256488400660177843021, 5.70932776538638318868350037135, 6.741247977508998978124357917, 7.26376451871454536371404427253, 7.8057013800415465653702233021, 8.889828575715591115062902851320, 9.80995705419524728047513833258, 10.57815591593098654314354520439, 11.38138022930488017664947168473, 12.0057145219225799869678324638, 13.335449457940522628616059898161, 13.87615756803871139170714358868, 14.43596913340016134460021679235, 15.73660844618798140955047271906, 16.31931268810046409544906932119, 17.01070203323261231596487560781, 17.566944034300565967500132189461, 18.24492419493736519480770991669, 19.08321523512594622142083371746, 19.52108848368839186680491051387, 20.56947205122614492104544283964

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