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Properties

Label 1-2240-2240.157-r0-0-0
Degree $1$
Conductor $2240$
Sign $0.827 - 0.561i$
Analytic cond. $10.4025$
Root an. cond. $10.4025$
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  + (0.991 − 0.130i)3-s + (0.965 − 0.258i)9-s + (0.793 + 0.608i)11-s + (−0.923 − 0.382i)13-s + (−0.5 − 0.866i)17-s + (0.608 + 0.793i)19-s + (−0.258 − 0.965i)23-s + (0.923 − 0.382i)27-s + (0.923 + 0.382i)29-s + (−0.5 − 0.866i)31-s + (0.866 + 0.5i)33-s + (0.130 − 0.991i)37-s + (−0.965 − 0.258i)39-s + (0.707 + 0.707i)41-s + (−0.382 − 0.923i)43-s + ⋯
L(s)  = 1  + (0.991 − 0.130i)3-s + (0.965 − 0.258i)9-s + (0.793 + 0.608i)11-s + (−0.923 − 0.382i)13-s + (−0.5 − 0.866i)17-s + (0.608 + 0.793i)19-s + (−0.258 − 0.965i)23-s + (0.923 − 0.382i)27-s + (0.923 + 0.382i)29-s + (−0.5 − 0.866i)31-s + (0.866 + 0.5i)33-s + (0.130 − 0.991i)37-s + (−0.965 − 0.258i)39-s + (0.707 + 0.707i)41-s + (−0.382 − 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.827 - 0.561i$
Analytic conductor: \(10.4025\)
Root analytic conductor: \(10.4025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2240,\ (0:\ ),\ 0.827 - 0.561i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.407034093 - 0.7392419638i\)
\(L(\frac12)\) \(\approx\) \(2.407034093 - 0.7392419638i\)
\(L(1)\) \(\approx\) \(1.544856197 - 0.1771288203i\)
\(L(1)\) \(\approx\) \(1.544856197 - 0.1771288203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.991 - 0.130i)T \)
11 \( 1 + (0.793 + 0.608i)T \)
13 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.608 + 0.793i)T \)
23 \( 1 + (-0.258 - 0.965i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.130 - 0.991i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.608 - 0.793i)T \)
59 \( 1 + (-0.608 + 0.793i)T \)
61 \( 1 + (0.793 - 0.608i)T \)
67 \( 1 + (0.991 - 0.130i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.965 + 0.258i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (0.923 + 0.382i)T \)
89 \( 1 + (-0.258 - 0.965i)T \)
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

−19.58174692949701878786272250906, −19.46778190112916634111284667037, −18.4453340732809268665069565555, −17.58927218777542929446104840590, −16.906501930162263690693435388599, −16.00340273965153202321475618470, −15.38751479053644360363129869095, −14.641082183493828199428407318222, −13.99775362399660895223256342812, −13.44945685945881276980497167643, −12.569490188677119379180310959491, −11.76274007269493391255582648875, −10.97333824739104700411556555761, −9.973306964786404182425210207283, −9.42131950580700809034682719649, −8.685568862840881909256319958635, −8.03613757865316582289551482506, −7.08843739621311542337607452552, −6.52693945017780725828635296785, −5.32696919526191526256176995776, −4.45498416152076214232787205953, −3.69837537744613822895724707650, −2.91382880228289326296723126872, −2.026529684019204925903417614959, −1.11613044422943773373582387475, 0.81145921564736435692872941309, 1.990972097880543446409556035469, 2.58201080378874640212145224619, 3.563303475116460989328823420552, 4.35593195971259771982722379380, 5.11494071645969470708930350946, 6.3493392575295410514307823548, 7.09910259086056808722492012166, 7.71780142247427578881844926140, 8.51039383067622033838321693310, 9.4502058951209199927884185847, 9.7484925943623649719061128845, 10.69940933398536943469134485347, 11.82713076126680555511718765657, 12.415608439475462861102989814524, 13.074087342696416681696494176883, 14.11965254698423564020133199789, 14.42542248730168478648020638117, 15.13230338225466113523148340711, 15.97131958306470567165356559707, 16.65437653881473462044382113732, 17.668275705930618367665596435638, 18.22514552909290068209950596051, 18.97855630521055838796560937792, 19.97025810365444783455296214686

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