Properties

Label 1-5520-5520.827-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.160 - 0.987i$
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  i·7-s i·11-s − 13-s + i·17-s + i·19-s i·29-s − 31-s + 37-s + 41-s + 43-s + i·47-s − 49-s + 53-s i·59-s + i·61-s + ⋯
L(s)  = 1  i·7-s i·11-s − 13-s + i·17-s + i·19-s i·29-s − 31-s + 37-s + 41-s + 43-s + i·47-s − 49-s + 53-s i·59-s + i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.160 - 0.987i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.160 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.064368896 - 0.9055690985i\)
\(L(\frac12)\) \(\approx\) \(1.064368896 - 0.9055690985i\)
\(L(1)\) \(\approx\) \(0.9771048760 - 0.1920980632i\)
\(L(1)\) \(\approx\) \(0.9771048760 - 0.1920980632i\)

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 \( 1 \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 - iT \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 - T \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 + iT \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 \)
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

−17.99763336393704750178447536531, −17.549732408522968697838802345502, −16.6254210067193605318871387580, −16.01951276673520916490803981851, −15.30172950233683124332493401418, −14.76743457963447954908958744548, −14.26651012562692926783452514190, −13.214457130332922859165209488403, −12.67813696464119487707977803852, −12.04998327907306750113624946472, −11.50846666690520387123474630879, −10.69257824147438888099382961471, −9.836042432698033132832904291404, −9.203899718245232037366193316158, −8.88710483915824136452797516289, −7.6528341756751863841589136135, −7.28751733975147384502316468982, −6.52333999805217403645370513108, −5.55120529885464500921629480923, −4.99356924866774167961725316809, −4.42955936130252210044785321448, −3.30847881031983878433931612863, −2.4087194760197514301172864794, −2.16112338763240121106388756204, −0.83105482115110356167559489464, 0.45103923418738635333808077402, 1.330755781659660659769933571547, 2.2774792374218492023786457251, 3.15455489431636583549992629458, 4.01669963966200254338731496696, 4.367193308472317210776111530382, 5.59129312968068533039661755926, 5.99225885228277885461191149972, 6.89800182679428796680456455525, 7.71815251903563519522354011112, 8.0413309205951870902541916184, 9.01997702471124866377529940553, 9.77935105253638572921531116777, 10.380243416047128896987803835117, 11.01105976027920719627611977616, 11.628300624944995289065062116040, 12.59363630355745337414656593800, 13.00049293155968445487822976159, 13.847833430644220978066005022422, 14.44274105292549798085426955048, 14.864738568214003496249472547305, 15.94335435727790206087361653130, 16.4533866918553438388315841718, 17.08410929874001623857386657999, 17.505301359332724237589422473786

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