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Properties

Label 1-240-240.83-r1-0-0
Degree $1$
Conductor $240$
Sign $-0.987 + 0.160i$
Analytic cond. $25.7915$
Root an. cond. $25.7915$
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 − i·11-s − 13-s + i·17-s + i·19-s − i·23-s + i·29-s − 31-s − 37-s + 41-s − 43-s − i·47-s − 49-s − 53-s + i·59-s + ⋯
L(s)  = 1  − i·7-s − i·11-s − 13-s + i·17-s + i·19-s − i·23-s + i·29-s − 31-s − 37-s + 41-s − 43-s − i·47-s − 49-s − 53-s + i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.987 + 0.160i$
Analytic conductor: \(25.7915\)
Root analytic conductor: \(25.7915\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 240,\ (1:\ ),\ -0.987 + 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01774295833 - 0.2201043593i\)
\(L(\frac12)\) \(\approx\) \(0.01774295833 - 0.2201043593i\)
\(L(1)\) \(\approx\) \(0.8006816524 - 0.1299327450i\)
\(L(1)\) \(\approx\) \(0.8006816524 - 0.1299327450i\)

Euler product

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

−26.39468146225972509785714739611, −25.38686756170531515177257897289, −24.75438136639864701880813427916, −23.73603470207056493742201265511, −22.59364366913973131402460651557, −21.945228524290550003250763136974, −20.92784548357680384550265895239, −19.90000965085716447624355569896, −19.03575926702200180079498079624, −17.93866639241588472281856820022, −17.29179090092328734275342797482, −15.87409173241923396878953210208, −15.22066547681737834824528860200, −14.24392359834627117801354145095, −12.97627782640295995054751797394, −12.10208499329078841158158557880, −11.264775504178352776325736930257, −9.72518156383206779424800679814, −9.21761117713792563211887966518, −7.76596298664072311559810045205, −6.856356250783098517912359124230, −5.44617472604229192354173661475, −4.621676038075325180016319171248, −2.93901747416417136651954290084, −1.93280413186258102059531705215, 0.067448963105514796776224935792, 1.56639126664496161295244757372, 3.2131894540678199562896326534, 4.24838288224369724719901116392, 5.56490631559548881805191998779, 6.750916539888534341115480506075, 7.7975221219391697813086167184, 8.82145582895739037561101673732, 10.21746276269951579529996333390, 10.79167885769729210289615557803, 12.12903911854059134985835873862, 13.07725030293458676183133630318, 14.173749301747250442467903146405, 14.84019857876283810923944235351, 16.42289947786992869914647172634, 16.776876929413941049784270925797, 17.95635781104975707976863681333, 19.09174756785500680579509143141, 19.84002141146182686254803893507, 20.82812575038601963451749580776, 21.8062794558502212993060215966, 22.69415485964946117528816395027, 23.799124490345096442243543840066, 24.35448956351720493959416707317, 25.53013506943827650354809614381

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