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Properties

Label 1-2640-2640.1949-r0-0-0
Degree $1$
Conductor $2640$
Sign $-0.967 + 0.254i$
Analytic cond. $12.2601$
Root an. cond. $12.2601$
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  + (−0.309 − 0.951i)7-s + (0.587 − 0.809i)13-s + (0.809 − 0.587i)17-s + (−0.951 − 0.309i)19-s − 23-s + (−0.951 + 0.309i)29-s + (−0.809 − 0.587i)31-s + (−0.951 + 0.309i)37-s + (−0.309 + 0.951i)41-s − i·43-s + (0.309 − 0.951i)47-s + (−0.809 + 0.587i)49-s + (−0.587 + 0.809i)53-s + (−0.951 + 0.309i)59-s + (0.587 + 0.809i)61-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)7-s + (0.587 − 0.809i)13-s + (0.809 − 0.587i)17-s + (−0.951 − 0.309i)19-s − 23-s + (−0.951 + 0.309i)29-s + (−0.809 − 0.587i)31-s + (−0.951 + 0.309i)37-s + (−0.309 + 0.951i)41-s − i·43-s + (0.309 − 0.951i)47-s + (−0.809 + 0.587i)49-s + (−0.587 + 0.809i)53-s + (−0.951 + 0.309i)59-s + (0.587 + 0.809i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2640\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.967 + 0.254i$
Analytic conductor: \(12.2601\)
Root analytic conductor: \(12.2601\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2640} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2640,\ (0:\ ),\ -0.967 + 0.254i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04538490443 - 0.3504095046i\)
\(L(\frac12)\) \(\approx\) \(-0.04538490443 - 0.3504095046i\)
\(L(1)\) \(\approx\) \(0.8127984407 - 0.1987242568i\)
\(L(1)\) \(\approx\) \(0.8127984407 - 0.1987242568i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.951 + 0.309i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (0.587 + 0.809i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−19.43685471767418312319729271622, −19.03755665098388143520684140482, −18.45370701623133123104871286168, −17.635071681630214002767027731786, −16.83425811827684255894933270954, −16.10941645351105628738809544665, −15.60143352406945990938024047536, −14.610822313610628907388862408861, −14.259778469679192125071800820372, −13.16088799214603826303458396622, −12.54771234400242556697119295338, −11.93806040369414987433573962504, −11.13388329054710286728340938435, −10.35102443253193196814829871809, −9.51738090978335022084966519212, −8.83495789274479972514194915007, −8.20826454106740009126529077084, −7.30470128518634364361096105849, −6.23985686096859190787049268085, −5.92687220649003317187666873240, −4.95258024900106885543403082732, −3.90988495187701105523010391453, −3.334818953852820443399390639735, −2.10164926963860368368029460015, −1.62730999442573302677956843259, 0.11037112674354328499486399535, 1.17662145282242443857006360163, 2.19982735133865170684278686652, 3.36441228999150487637924692304, 3.80421472365908586054373287688, 4.813831454010342380552709014750, 5.69289317225800624922658099697, 6.42985284204505611284601050204, 7.33600838079675334281981577620, 7.87122511921742495871479349044, 8.77815280385137602796110802520, 9.62149222072576527305810306668, 10.42046620577756592003776833238, 10.825510253035632326971340770032, 11.8114645605804359440678426924, 12.5864918364205898069033162464, 13.35782114082810803249406303484, 13.81090318824885731685003651418, 14.73471638032473042045559720713, 15.38643894881698122728052276734, 16.27501294971261563093623363993, 16.77388043272865152657587719732, 17.48195960730733978434259323261, 18.31348369212310551770709165564, 18.88291348628894059610978523358

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