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Properties

Label 1-1480-1480.117-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.309 + 0.950i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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·3-s + i·7-s − 9-s + 11-s − 13-s − 17-s + i·19-s + 21-s + 23-s + i·27-s − i·29-s − i·31-s − i·33-s + i·39-s − 41-s + ⋯
L(s)  = 1  − i·3-s + i·7-s − 9-s + 11-s − 13-s − 17-s + i·19-s + 21-s + 23-s + i·27-s − i·29-s − i·31-s − i·33-s + i·39-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.309 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7181730496 + 0.5216522175i\)
\(L(\frac12)\) \(\approx\) \(0.7181730496 + 0.5216522175i\)
\(L(1)\) \(\approx\) \(0.8996040832 - 0.05824770417i\)
\(L(1)\) \(\approx\) \(0.8996040832 - 0.05824770417i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 + iT \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
41 \( 1 \)
43 \( 1 - T \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - T \)
67 \( 1 \)
71 \( 1 + iT \)
73 \( 1 \)
79 \( 1 + T \)
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

−20.289209234560985588708239032347, −19.91577821705690942453192545546, −19.41879132853492975907998100224, −18.03755983629778812079339313731, −17.09145031352236385892270696133, −17.01306033208282744463540742889, −16.01514487612888661056594772932, −15.17923711266421039873298150229, −14.55557041577183626545418240278, −13.85321573732105008346949002443, −13.00965294612544614874503207496, −11.9178892748237084732125278208, −11.14612604429285737620559084313, −10.571335107821856881026356768674, −9.6855017452784518699764673910, −9.07527244041938778692169392791, −8.278008922023191766599338934147, −6.89529262891162195828834202038, −6.714377950755589492281188030975, −5.019713264362949239020488508045, −4.805247328716813584557150532200, −3.729431721395075857244903811170, −3.06699035298206642478865930838, −1.77984203780408482496017328939, −0.33672052823532552785910998078, 1.23543395143295563636360944963, 2.18108067456221748107308445478, 2.82670130605326107130133926426, 4.082215798488823225091282563551, 5.177933648045775162017485015643, 6.03543546460296508838544016017, 6.673943827777184236956616666493, 7.52579970315436002221282297142, 8.412572887134710897378242159853, 9.08291825561071021468784856146, 9.8680012907240661210111610872, 11.19117900783210211492498966697, 11.80798568533725934403507658550, 12.34623876814484202600003623203, 13.139422236161298046296311844861, 13.93457148339866136762388657759, 14.86794478188411995141723016807, 15.19995154767545614070561578184, 16.56330243077461702108936073673, 17.162715060713273720337061552455, 17.82475423457043649298473600868, 18.74910906100079451104150254146, 19.13865273110543152367162435499, 19.88733503579399095205669991982, 20.67550012829812701526603915769

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