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Properties

Label 1-5520-5520.2957-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.377 - 0.926i$
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  + (0.540 − 0.841i)7-s + (0.989 − 0.142i)11-s + (0.841 − 0.540i)13-s + (0.281 + 0.959i)17-s + (−0.281 + 0.959i)19-s + (−0.281 − 0.959i)29-s + (0.415 − 0.909i)31-s + (−0.654 + 0.755i)37-s + (−0.654 − 0.755i)41-s + (−0.415 − 0.909i)43-s − i·47-s + (−0.415 − 0.909i)49-s + (0.841 + 0.540i)53-s + (−0.540 − 0.841i)59-s + (0.909 + 0.415i)61-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)7-s + (0.989 − 0.142i)11-s + (0.841 − 0.540i)13-s + (0.281 + 0.959i)17-s + (−0.281 + 0.959i)19-s + (−0.281 − 0.959i)29-s + (0.415 − 0.909i)31-s + (−0.654 + 0.755i)37-s + (−0.654 − 0.755i)41-s + (−0.415 − 0.909i)43-s − i·47-s + (−0.415 − 0.909i)49-s + (0.841 + 0.540i)53-s + (−0.540 − 0.841i)59-s + (0.909 + 0.415i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.377 - 0.926i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (2957, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.377 - 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.759601627 - 1.183184799i\)
\(L(\frac12)\) \(\approx\) \(1.759601627 - 1.183184799i\)
\(L(1)\) \(\approx\) \(1.233073564 - 0.2363316302i\)
\(L(1)\) \(\approx\) \(1.233073564 - 0.2363316302i\)

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 \( 1 + (0.540 - 0.841i)T \)
11 \( 1 + (0.989 - 0.142i)T \)
13 \( 1 + (0.841 - 0.540i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (-0.281 + 0.959i)T \)
29 \( 1 + (-0.281 - 0.959i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (-0.654 + 0.755i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (-0.540 - 0.841i)T \)
61 \( 1 + (0.909 + 0.415i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (0.281 - 0.959i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (0.755 - 0.654i)T \)
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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

−17.90209304500206370631885561633, −17.561037783136092447625899753820, −16.51391503435080828606755609061, −16.10031831245630122048869092621, −15.33587199327356523849144814290, −14.633365178166031732314206093604, −14.14687793106499341781220874547, −13.43498977392504401612906465559, −12.59577512851960818530917405713, −11.90740198173436725137178569565, −11.40574578308105614661090864149, −10.862948891462458769478431434099, −9.82555429132217484749505365834, −9.04008382019918069852790210049, −8.80597659429078856523045726729, −7.95610139462157185359462418343, −6.93252985746441932929195292056, −6.58057568652974194884232858938, −5.6156495922628779127074172733, −4.9563790377495983159578105781, −4.279541201066925054434634290978, −3.358953147858514452727908777654, −2.62578590280179833333053766580, −1.68298969161472923920567058034, −1.06829162254369453627797797828, 0.596989493927925577158778234983, 1.47257376656174788591655147619, 2.05158508558356160700452443937, 3.515053715862766789854996419213, 3.7493329928480888868516794747, 4.50176688565969198766379312322, 5.53753490112978184441837254104, 6.13062543286058632392410376992, 6.83562542379863042109202750445, 7.68272055864340952886036156395, 8.33475366871942424009683682485, 8.78085043485712885743887713089, 9.989990855545223587700667706632, 10.27586802631412011055368692191, 11.086585390410320925791895466474, 11.7328904955799438613712405695, 12.35329378728973005543962263325, 13.327132507667447512575891284692, 13.697143967000324286833411510187, 14.46404315938561348037125883931, 15.06202282381894247657947620697, 15.6749408762095958900890968872, 16.74324010133716006552745351817, 17.033031599512867426318703796337, 17.504519952770787585800236885565

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