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Properties

Label 1-2340-2340.1939-r0-0-0
Degree $1$
Conductor $2340$
Sign $0.622 - 0.782i$
Analytic cond. $10.8669$
Root an. cond. $10.8669$
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·7-s + (−0.866 + 0.5i)11-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s − 23-s + (−0.5 − 0.866i)29-s + (0.866 − 0.5i)31-s + (0.866 + 0.5i)37-s − i·41-s − 43-s + (−0.866 − 0.5i)47-s − 49-s − 53-s + (0.866 + 0.5i)59-s + 61-s + ⋯
L(s)  = 1  − i·7-s + (−0.866 + 0.5i)11-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s − 23-s + (−0.5 − 0.866i)29-s + (0.866 − 0.5i)31-s + (0.866 + 0.5i)37-s − i·41-s − 43-s + (−0.866 − 0.5i)47-s − 49-s − 53-s + (0.866 + 0.5i)59-s + 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.622 - 0.782i$
Analytic conductor: \(10.8669\)
Root analytic conductor: \(10.8669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2340,\ (0:\ ),\ 0.622 - 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010175330 - 0.4870080324i\)
\(L(\frac12)\) \(\approx\) \(1.010175330 - 0.4870080324i\)
\(L(1)\) \(\approx\) \(0.9395400542 + 0.003128590278i\)
\(L(1)\) \(\approx\) \(0.9395400542 + 0.003128590278i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 - iT \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 - iT \)
43 \( 1 - T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 - T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + T \)
67 \( 1 - iT \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + iT \)
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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

−19.80055707753289924938651930464, −19.03650163506703910908536268832, −18.123049459614980139430968879971, −17.70465239564166331139925806888, −16.695553300179463621313832858, −16.222040966820014884818650240115, −15.5248549790668354371152315268, −14.48407327821196617075297897313, −14.00669919919568219630947888772, −13.12019468877559511316838123733, −12.71270435629092864689003197812, −11.5126324082890160277053037672, −11.00058856896579354441631267445, −10.12440028550504002127448821268, −9.71216433637242291336124178913, −8.35283062598798159224149338375, −8.041200956517649918330797600124, −7.100641094984877158190784516283, −6.316413111783715728824708726052, −5.46811945022744182890617190173, −4.600782755839603907451032889319, −3.744740989486137680268158164990, −3.03850922902854231022295948400, −1.88534830178987355930855699564, −0.93719679104068268355302262336, 0.42270212057992065910239195011, 1.96976296032781441891600368567, 2.50535580289723362851813056729, 3.40074701260733304331916706558, 4.61413018700680981475557965017, 5.17086376967159213861054955594, 6.00244759087856292268423427471, 6.83180156432638977015607172199, 7.79853088128471977858372779389, 8.31239432978145611083987151080, 9.45830821839216340298913617406, 9.71718276338804385419231725206, 10.78896399057488186922417122518, 11.75290334603827287086394608627, 11.99474212667963132168210309052, 13.15655464278065175963757712671, 13.533330355870109425873391331325, 14.56058177780909290388245721263, 15.39205774572891918476385142509, 15.729211721976364374720639793, 16.50355819581820263940130864741, 17.59153377642550988233397088133, 18.11327405268267794487418043656, 18.623743307893414039479448552927, 19.46576676311757952002437424049

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