↑

Properties

Label 1-2952-2952.1267-r1-0-0
Degree $1$
Conductor $2952$
Sign $-0.867 + 0.497i$
Analytic cond. $317.236$
Root an. cond. $317.236$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.913 + 0.406i)5-s + (−0.669 + 0.743i)7-s + (0.913 + 0.406i)11-s + (−0.669 − 0.743i)13-s + (−0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.669 − 0.743i)23-s + (0.669 − 0.743i)25-s + (0.104 − 0.994i)29-s + (0.104 + 0.994i)31-s + (0.309 − 0.951i)35-s + (0.809 + 0.587i)37-s + (−0.978 + 0.207i)43-s + (0.978 − 0.207i)47-s + (−0.104 − 0.994i)49-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)5-s + (−0.669 + 0.743i)7-s + (0.913 + 0.406i)11-s + (−0.669 − 0.743i)13-s + (−0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.669 − 0.743i)23-s + (0.669 − 0.743i)25-s + (0.104 − 0.994i)29-s + (0.104 + 0.994i)31-s + (0.309 − 0.951i)35-s + (0.809 + 0.587i)37-s + (−0.978 + 0.207i)43-s + (0.978 − 0.207i)47-s + (−0.104 − 0.994i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2952\)    =    \(2^{3} \cdot 3^{2} \cdot 41\)
Sign: $-0.867 + 0.497i$
Analytic conductor: \(317.236\)
Root analytic conductor: \(317.236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2952} (1267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2952,\ (1:\ ),\ -0.867 + 0.497i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2643295434 + 0.9927411564i\)
\(L(\frac12)\) \(\approx\) \(0.2643295434 + 0.9927411564i\)
\(L(1)\) \(\approx\) \(0.7789743157 + 0.2211250594i\)
\(L(1)\) \(\approx\) \(0.7789743157 + 0.2211250594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (-0.913 + 0.406i)T \)
7 \( 1 + (-0.669 + 0.743i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.669 - 0.743i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (0.104 - 0.994i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (-0.978 + 0.207i)T \)
47 \( 1 + (0.978 - 0.207i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.8388172179296432728124050048, −17.88922453753754397597308636678, −16.99260288915516219968537035800, −16.59759930914097117857797205496, −15.885164039290682028840899692288, −15.29579721292881856350546031742, −14.31064968465663879249891443907, −13.7302845146688721849946658734, −12.984711976781509791120219081765, −12.21969651334957662357502052487, −11.389532056333724120107091105164, −11.16334721856359725899157142263, −9.86319630906611812767738259378, −9.318364106267671651473815574800, −8.67905916501890479001648711763, −7.657959655900354402923809777490, −7.01619962176522813660050784371, −6.53344013849947642667370180523, −5.33280939617373192968788562405, −4.443801640578836847522181293330, −3.92634707560419467784886357734, −3.16633357835989097666401341106, −2.06688506009114964730195782086, −0.84964528736590218410699580811, −0.25890561557211275257359212883, 0.80820267282068550589196498820, 2.11869771197935873600153009484, 2.831066608046028583538212982030, 3.74391019924849094478421091791, 4.31579181620574688218230978335, 5.359651946460352255782442012611, 6.33821775900427065800154082034, 6.73968237351714340023822633512, 7.780243953577280581309650996536, 8.34404895587802766327006946301, 9.177572153687055000015758609291, 10.02972446659307264211812058349, 10.56801804334528988827856172888, 11.66241090985536184150457963544, 12.14793940913884198618608535359, 12.58292590961135485391011056514, 13.59106492214570250386529461053, 14.54353479304013191502253106714, 15.09850810918055336951241300406, 15.541051533402259452998870173356, 16.4028173417572404725374199261, 17.02490596087392726414985307863, 17.997166656247004547997286246828, 18.523031477499325321004221301811, 19.316852444568759080523625911375

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