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Properties

Label 1-5520-5520.803-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.996 - 0.0835i$
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  + (0.989 − 0.142i)7-s + (0.909 + 0.415i)11-s + (0.142 − 0.989i)13-s + (−0.755 + 0.654i)17-s + (0.755 + 0.654i)19-s + (0.755 − 0.654i)29-s + (0.959 + 0.281i)31-s + (0.841 − 0.540i)37-s + (0.841 + 0.540i)41-s + (−0.959 + 0.281i)43-s − i·47-s + (0.959 − 0.281i)49-s + (−0.142 − 0.989i)53-s + (−0.989 − 0.142i)59-s + (−0.281 + 0.959i)61-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)7-s + (0.909 + 0.415i)11-s + (0.142 − 0.989i)13-s + (−0.755 + 0.654i)17-s + (0.755 + 0.654i)19-s + (0.755 − 0.654i)29-s + (0.959 + 0.281i)31-s + (0.841 − 0.540i)37-s + (0.841 + 0.540i)41-s + (−0.959 + 0.281i)43-s − i·47-s + (0.959 − 0.281i)49-s + (−0.142 − 0.989i)53-s + (−0.989 − 0.142i)59-s + (−0.281 + 0.959i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.996 - 0.0835i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.996 - 0.0835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.451573251 - 0.1025590756i\)
\(L(\frac12)\) \(\approx\) \(2.451573251 - 0.1025590756i\)
\(L(1)\) \(\approx\) \(1.351422463 + 0.02172851320i\)
\(L(1)\) \(\approx\) \(1.351422463 + 0.02172851320i\)

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

−17.93343785579466731109301230466, −17.182725602159419726584681618401, −16.68218920442832334510647154141, −15.80899668263883944426651351758, −15.33417232608230256096332384358, −14.39664481826584173512096009248, −13.94111688883862856464711582128, −13.54859151444640791716413874968, −12.362449949367529066425694712794, −11.814086601256347259780592041058, −11.231227229883547734991685678989, −10.82342224752849841756706567116, −9.56971142952999464809891813817, −9.207405546404056109001310170976, −8.48508156893528626296225426476, −7.79598266817059339775733078897, −6.89656611920782650890502987947, −6.440722444907206775632149739661, −5.51414632691846919625053866773, −4.577718698967235911637487669544, −4.37263769936608630202358266908, −3.208582633405112632633953692414, −2.456240182713058216087278901403, −1.53811409552811283129956089756, −0.87402855555834840068278472958, 0.84350838240743202873131760485, 1.53133609294775516427416526313, 2.36198665401490354034694884478, 3.30968724408002898753769062544, 4.1581367828084135497501519515, 4.68375537806985377104227432021, 5.549785592144252825053712791969, 6.26157019315134152435613014025, 6.99141222957469316855733555388, 7.95281722731902249244100888578, 8.204358520405400071042052666526, 9.09852548566928314400521248452, 9.91135327150034074747231987248, 10.4928888165581529867176822407, 11.26544485841028307923177778769, 11.847981567801820345268394151583, 12.45132281097824615961759687570, 13.34267096346761546254562034098, 13.87370337452225354332778682679, 14.80043156435708833556708174717, 14.9578932822698077919172233739, 15.865518466003209646388037145735, 16.59652205431032084663423447751, 17.425172522211171448432146053673, 17.74096936834811311045600845241

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