Properties

Label 1-240-240.59-r0-0-0
Degree $1$
Conductor $240$
Sign $0.382 + 0.923i$
Analytic cond. $1.11455$
Root an. cond. $1.11455$
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  − 7-s + i·11-s + i·13-s + 17-s + i·19-s + 23-s + i·29-s − 31-s i·37-s + 41-s + i·43-s − 47-s + 49-s + i·53-s + i·59-s + ⋯
L(s)  = 1  − 7-s + i·11-s + i·13-s + 17-s + i·19-s + 23-s + i·29-s − 31-s i·37-s + 41-s + i·43-s − 47-s + 49-s + i·53-s + i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(1.11455\)
Root analytic conductor: \(1.11455\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 240,\ (0:\ ),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8179913775 + 0.5465643644i\)
\(L(\frac12)\) \(\approx\) \(0.8179913775 + 0.5465643644i\)
\(L(1)\) \(\approx\) \(0.9255553623 + 0.2068141687i\)
\(L(1)\) \(\approx\) \(0.9255553623 + 0.2068141687i\)

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 \( 1 - T \)
23 \( 1 \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + T \)
67 \( 1 \)
71 \( 1 + iT \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 + T \)
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

−25.9311018812890220116832208078, −25.21069262145087570384247973178, −24.194305218230777763329711252246, −23.16316984729890701729851397392, −22.377658215580481820578206361346, −21.475318299398398128156326144500, −20.420248273828803752057179682655, −19.36960133500664622071692062355, −18.78156681358589885767477273910, −17.51762032108246927399050942233, −16.56075638110085335563334464985, −15.733970600914965541872646642716, −14.73186433173428841512812842450, −13.43694810752523968771364938888, −12.86096671933584119990289189374, −11.602186668077718863957088870425, −10.54506955275824459249650229629, −9.56160650572696755496255185412, −8.50568310426989267554989362471, −7.327438240311009893739750044791, −6.16549271699470535027648507755, −5.231993443398095803011504825059, −3.57518561733692363749664924493, −2.78557517359073033875421664225, −0.739849064957020356426768335863, 1.60847144595022416460358667338, 3.08835759373921316614357212660, 4.21582448339759127435915467088, 5.56172289228254004552301182437, 6.72917205762816271041628317393, 7.59070552961348869572795779873, 9.12411298931315809830848283879, 9.78023560555355267432743021397, 10.914312953723082174110329258038, 12.280424435601826505470368736449, 12.79506717404215774686839534226, 14.14817469067739343819335936206, 14.94183823268132981244966016346, 16.251685411371243413094899115033, 16.72994815093635294046256458765, 18.072175547236951136101345203296, 18.96454883992551183042288324927, 19.77820810877779868925675222714, 20.84517494035208035922719620924, 21.717512542393328459808081755026, 22.94778824093387598606277580794, 23.269205593597592849126078812468, 24.65985991232625457672635792814, 25.57397055034247572071992023905, 26.15190749765898726681574360401

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