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Properties

Label 1-5520-5520.2627-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.923 + 0.382i$
Analytic cond. $25.6347$
Root an. cond. $25.6347$
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  + (0.755 + 0.654i)7-s + (0.540 + 0.841i)11-s + (−0.654 − 0.755i)13-s + (0.909 + 0.415i)17-s + (0.909 − 0.415i)19-s + (0.909 + 0.415i)29-s + (0.142 − 0.989i)31-s + (0.959 − 0.281i)37-s + (−0.959 − 0.281i)41-s + (0.142 + 0.989i)43-s − i·47-s + (0.142 + 0.989i)49-s + (0.654 − 0.755i)53-s + (0.755 − 0.654i)59-s + (0.989 + 0.142i)61-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)7-s + (0.540 + 0.841i)11-s + (−0.654 − 0.755i)13-s + (0.909 + 0.415i)17-s + (0.909 − 0.415i)19-s + (0.909 + 0.415i)29-s + (0.142 − 0.989i)31-s + (0.959 − 0.281i)37-s + (−0.959 − 0.281i)41-s + (0.142 + 0.989i)43-s − i·47-s + (0.142 + 0.989i)49-s + (0.654 − 0.755i)53-s + (0.755 − 0.654i)59-s + (0.989 + 0.142i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (2627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.293776598 + 0.4560091575i\)
\(L(\frac12)\) \(\approx\) \(2.293776598 + 0.4560091575i\)
\(L(1)\) \(\approx\) \(1.299165982 + 0.1263564793i\)
\(L(1)\) \(\approx\) \(1.299165982 + 0.1263564793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (0.755 + 0.654i)T \)
11 \( 1 + (0.540 + 0.841i)T \)
13 \( 1 + (-0.654 - 0.755i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (0.909 - 0.415i)T \)
29 \( 1 + (0.909 + 0.415i)T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 + (0.959 - 0.281i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (0.755 - 0.654i)T \)
61 \( 1 + (0.989 + 0.142i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (-0.909 + 0.415i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.281 + 0.959i)T \)
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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

−17.825262990209692554166457943720, −17.06441457136442861565210561855, −16.52333208389597040200646287469, −16.11611613894238193346171652398, −14.979657450035801564725806302298, −14.48370027682603639233730930393, −13.788015390969919522430374728605, −13.57908465299690809702930528049, −12.27059198439488841754252567234, −11.76722997359009789771173232968, −11.39348260900985273927035176989, −10.25791156472255902984523644190, −10.03778881117458728222582065497, −8.97021856486432561323389071892, −8.44079892493181662185067800695, −7.57183488233977335131391316975, −7.108779537912936247672115550915, −6.26750932427186039371351301260, −5.39279931354858928392398091381, −4.80168685175491603783136598407, −3.97286180818300353582659091018, −3.31389228147485316171395988544, −2.406475384616871194764475571179, −1.36473102173074151699564398663, −0.82890048979959042542526663995, 0.85781518179474707231463181090, 1.666509191014237132037905159955, 2.519760494114448061044745946940, 3.19027754948772314783412046815, 4.23035160439228139872396694878, 4.91220133831167551500788679126, 5.495966765984391871626587642721, 6.25351021462751440023333566541, 7.200646067119118612602748999077, 7.79001219415161858060541467473, 8.362126552611450172631603344644, 9.295619035849587700799066133556, 9.814229549040760583555161808900, 10.4835760567751433846137752892, 11.46642746476431393716873576465, 11.88273859680119319010180729119, 12.53481130737043508118273849205, 13.15977904072054148708671884810, 14.17037687688260220156675053434, 14.70769143591316394108862752229, 15.0944350125226644852227486067, 15.87631041963558676255872668772, 16.637321944510748738907104057772, 17.45903224162168152965162286950, 17.80159208028836887055301801807

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