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Properties

Label 1-2652-2652.1043-r1-0-0
Degree $1$
Conductor $2652$
Sign $-0.840 + 0.541i$
Analytic cond. $284.996$
Root an. cond. $284.996$
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  + (0.382 + 0.923i)5-s + (−0.991 + 0.130i)7-s + (0.793 − 0.608i)11-s + (−0.258 + 0.965i)19-s + (−0.793 + 0.608i)23-s + (−0.707 + 0.707i)25-s + (0.991 + 0.130i)29-s + (0.923 − 0.382i)31-s + (−0.5 − 0.866i)35-s + (−0.130 + 0.991i)37-s + (0.608 + 0.793i)41-s + (0.965 + 0.258i)43-s − i·47-s + (0.965 − 0.258i)49-s + (−0.707 − 0.707i)53-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)5-s + (−0.991 + 0.130i)7-s + (0.793 − 0.608i)11-s + (−0.258 + 0.965i)19-s + (−0.793 + 0.608i)23-s + (−0.707 + 0.707i)25-s + (0.991 + 0.130i)29-s + (0.923 − 0.382i)31-s + (−0.5 − 0.866i)35-s + (−0.130 + 0.991i)37-s + (0.608 + 0.793i)41-s + (0.965 + 0.258i)43-s − i·47-s + (0.965 − 0.258i)49-s + (−0.707 − 0.707i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2652\)    =    \(2^{2} \cdot 3 \cdot 13 \cdot 17\)
Sign: $-0.840 + 0.541i$
Analytic conductor: \(284.996\)
Root analytic conductor: \(284.996\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2652} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2652,\ (1:\ ),\ -0.840 + 0.541i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4197487021 + 1.426034953i\)
\(L(\frac12)\) \(\approx\) \(0.4197487021 + 1.426034953i\)
\(L(1)\) \(\approx\) \(0.9781229660 + 0.2944174048i\)
\(L(1)\) \(\approx\) \(0.9781229660 + 0.2944174048i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
17 \( 1 \)
good5 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 + (-0.991 + 0.130i)T \)
11 \( 1 + (0.793 - 0.608i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (-0.793 + 0.608i)T \)
29 \( 1 + (0.991 + 0.130i)T \)
31 \( 1 + (0.923 - 0.382i)T \)
37 \( 1 + (-0.130 + 0.991i)T \)
41 \( 1 + (0.608 + 0.793i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.965 + 0.258i)T \)
61 \( 1 + (-0.991 + 0.130i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.793 - 0.608i)T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.608 - 0.793i)T \)
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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

−19.14943529017215171461144769844, −17.890414995226999851841651596233, −17.44014936462869014162020454061, −16.81181523637003677543925566394, −15.867935323136193317586930911429, −15.716487838625904533900592276366, −14.41606287459808644454378074568, −13.87013094788824451603987547103, −13.07333552842349183446737426963, −12.36977872078218512177664006502, −12.05038015720129504079313498598, −10.81431427876218324353893079927, −10.08393691010274737231012777819, −9.31150201930935789290712863495, −8.94981979594930409331845080728, −7.97861648026047582147654303494, −6.98272937810770310833551974986, −6.339480715991104721545028656632, −5.644336883992699243324393679910, −4.483398567978237712978230427161, −4.187632989910046573629909650210, −2.93091932124531220587989698504, −2.126680187970077188717698079347, −1.04731265176656922548235883934, −0.28249288749659633628818599770, 0.992222098972732578481186904568, 2.049803342180077873125766046585, 3.00109629395805215679271735460, 3.52525004831393134750533593024, 4.41767641709613423481008905698, 5.7898361197320845564785712582, 6.18769424450956338784043968657, 6.76860306628629709434896811092, 7.728502092256102604518964926013, 8.572703087904781246984182780091, 9.50812934511415463587915894706, 10.027580083603000217550538664938, 10.664434855394660953292352571557, 11.68207333628929870774928714480, 12.11691377516409633922466903495, 13.21577994691050521459921349799, 13.74884492294468926904873839064, 14.44013129598091177913678021121, 15.119094198059400950577609856169, 15.98834607109985387057871318193, 16.53622971898261652948973729025, 17.395720247243251009516728484329, 18.03303376022229965218950356243, 18.93433436891792240633257629820, 19.248160004155800182192620928076

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