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Properties

Label 1-2652-2652.1631-r1-0-0
Degree $1$
Conductor $2652$
Sign $0.884 + 0.466i$
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  − i·5-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)19-s + (−0.5 − 0.866i)23-s − 25-s + (−0.5 − 0.866i)29-s − i·31-s + (−0.5 + 0.866i)35-s + (−0.866 + 0.5i)37-s + (−0.866 + 0.5i)41-s + (−0.5 + 0.866i)43-s − i·47-s + (0.5 + 0.866i)49-s − 53-s + ⋯
L(s)  = 1  − i·5-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)19-s + (−0.5 − 0.866i)23-s − 25-s + (−0.5 − 0.866i)29-s − i·31-s + (−0.5 + 0.866i)35-s + (−0.866 + 0.5i)37-s + (−0.866 + 0.5i)41-s + (−0.5 + 0.866i)43-s − i·47-s + (0.5 + 0.866i)49-s − 53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2652\)    =    \(2^{2} \cdot 3 \cdot 13 \cdot 17\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(284.996\)
Root analytic conductor: \(284.996\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2652} (1631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2652,\ (1:\ ),\ 0.884 + 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1273757423 + 0.03150767632i\)
\(L(\frac12)\) \(\approx\) \(0.1273757423 + 0.03150767632i\)
\(L(1)\) \(\approx\) \(0.6377677559 - 0.2241477464i\)
\(L(1)\) \(\approx\) \(0.6377677559 - 0.2241477464i\)

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 \( 1 - iT \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 - T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−19.03598783211132720656422091714, −18.4856223166689899091956836740, −17.88140407013283786872871194426, −16.956795934109501699423216317883, −16.11098977075124203945665810637, −15.55803283832662125052819765803, −14.95828849614637784908059129573, −14.06360849508790075853853038134, −13.4770478297807777506986131948, −12.64110021721731380691242882725, −11.98961908135764115107287906055, −11.05964542360714169578646331131, −10.40413544528599377601133687320, −9.91063325534940022523242471984, −8.87407091401358117237089734179, −8.23278013153807976862713976906, −7.19213509368478885882225792943, −6.70406806735979274675018694738, −5.75408677408281884252314056785, −5.30616953688067566464957837394, −3.82524408033559108559512777124, −3.33225992868683156830710836573, −2.52223339272559992728091976494, −1.69972470408218274119278270470, −0.04633080740245479219173090132, 0.378899752259786038372800975411, 1.643931643964609417870451834792, 2.506054995029136225328787255299, 3.52117341795938199737646929926, 4.46346853966606477845456140731, 4.906124400844452498417865529998, 6.04613261160623426984269741824, 6.58534122643867028908908494677, 7.716101204020324349443706061052, 8.165763928856318307930450509593, 9.17379275939920786150135347122, 9.77440133565256586567440523155, 10.42341008711838445029892100415, 11.31215726135162884245344606332, 12.27349247945933785488183616201, 12.889932758503343365418661299541, 13.26358400602736886284680667762, 14.07372924166354151833734320045, 15.212863341297644529291259286169, 15.65763105008117769950613954103, 16.47924309162762791395780326922, 16.9848072239542364471262717145, 17.634339053155385328630564735363, 18.61346275350798910626516676712, 19.24305029726219367969713489334

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