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Properties

Label 1-4160-4160.1309-r0-0-0
Degree $1$
Conductor $4160$
Sign $-0.156 + 0.987i$
Analytic cond. $19.3189$
Root an. cond. $19.3189$
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  + (0.130 + 0.991i)3-s + (−0.258 − 0.965i)7-s + (−0.965 + 0.258i)9-s + (0.793 + 0.608i)11-s + (0.866 + 0.5i)17-s + (0.608 + 0.793i)19-s + (0.923 − 0.382i)21-s + (0.258 − 0.965i)23-s + (−0.382 − 0.923i)27-s + (0.130 + 0.991i)29-s − 31-s + (−0.5 + 0.866i)33-s + (−0.608 + 0.793i)37-s + (−0.258 + 0.965i)41-s + (0.130 − 0.991i)43-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)3-s + (−0.258 − 0.965i)7-s + (−0.965 + 0.258i)9-s + (0.793 + 0.608i)11-s + (0.866 + 0.5i)17-s + (0.608 + 0.793i)19-s + (0.923 − 0.382i)21-s + (0.258 − 0.965i)23-s + (−0.382 − 0.923i)27-s + (0.130 + 0.991i)29-s − 31-s + (−0.5 + 0.866i)33-s + (−0.608 + 0.793i)37-s + (−0.258 + 0.965i)41-s + (0.130 − 0.991i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4160\)    =    \(2^{6} \cdot 5 \cdot 13\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(19.3189\)
Root analytic conductor: \(19.3189\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4160} (1309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4160,\ (0:\ ),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.088532784 + 1.274679964i\)
\(L(\frac12)\) \(\approx\) \(1.088532784 + 1.274679964i\)
\(L(1)\) \(\approx\) \(1.042884870 + 0.3939303784i\)
\(L(1)\) \(\approx\) \(1.042884870 + 0.3939303784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.130 + 0.991i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
11 \( 1 + (0.793 + 0.608i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.608 + 0.793i)T \)
23 \( 1 + (0.258 - 0.965i)T \)
29 \( 1 + (0.130 + 0.991i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.608 + 0.793i)T \)
41 \( 1 + (-0.258 + 0.965i)T \)
43 \( 1 + (0.130 - 0.991i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.923 - 0.382i)T \)
59 \( 1 + (0.991 + 0.130i)T \)
61 \( 1 + (-0.793 + 0.608i)T \)
67 \( 1 + (0.130 + 0.991i)T \)
71 \( 1 + (0.258 + 0.965i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (0.965 + 0.258i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−18.37868264460628191952272538643, −17.598497292227095526521707604446, −17.04643976595992302385645347977, −16.158363936691508582472598784720, −15.54407092414557225822895293244, −14.66487633427143479626887628387, −14.09151803153940394828767591708, −13.46443187527037751621096555040, −12.74046009445486350796537594440, −12.02020459555599496197946452308, −11.605247452552408727247969310373, −10.94545088934014269655259513798, −9.61783807338565237719812531117, −9.182752940469619014259345269624, −8.562013932197875593147986293414, −7.64178056975743002036114365814, −7.144610097005850262129367946810, −6.20779644399144726618888186086, −5.72036815534438936506172431659, −5.03235503461998334681178974030, −3.65301561398630459478940717724, −3.094733002883801923496883483896, −2.28988721187889294714811253532, −1.441138468253540276666479773934, −0.53513083821139811994291270920, 0.970606566304107172212808765869, 1.893388123534508874285964598219, 3.14443781443854868563794364099, 3.65714389184963793200009395663, 4.272243415234016053548509238905, 5.07953364296445219714292177536, 5.81954533352078099117363828287, 6.78732924497496883441057433802, 7.37779318416025539306417148230, 8.359296218892889961213149015602, 8.93571888663188678423481605961, 9.94213122279570987523931833983, 10.13087946775405581945635075669, 10.83925164100991446606376018106, 11.74727697923062015535705195324, 12.34045946878411598084139291365, 13.24853864517920886461283610327, 14.055307590000145729400346980602, 14.64418662342616608321676209391, 14.986062370364604242468463403280, 16.11301475274415022502349734981, 16.58986214466037776908129123112, 16.94299457303627081579467550359, 17.76058465449756409619061307533, 18.63375049601856548002886961073

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