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Properties

Label 1-5520-5520.2963-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.286 - 0.958i$
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  + (−0.281 − 0.959i)7-s + (0.755 − 0.654i)11-s + (0.959 + 0.281i)13-s + (0.989 + 0.142i)17-s + (−0.989 + 0.142i)19-s + (−0.989 − 0.142i)29-s + (−0.841 − 0.540i)31-s + (0.415 − 0.909i)37-s + (0.415 + 0.909i)41-s + (0.841 − 0.540i)43-s − i·47-s + (−0.841 + 0.540i)49-s + (−0.959 + 0.281i)53-s + (0.281 − 0.959i)59-s + (0.540 − 0.841i)61-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)7-s + (0.755 − 0.654i)11-s + (0.959 + 0.281i)13-s + (0.989 + 0.142i)17-s + (−0.989 + 0.142i)19-s + (−0.989 − 0.142i)29-s + (−0.841 − 0.540i)31-s + (0.415 − 0.909i)37-s + (0.415 + 0.909i)41-s + (0.841 − 0.540i)43-s − i·47-s + (−0.841 + 0.540i)49-s + (−0.959 + 0.281i)53-s + (0.281 − 0.959i)59-s + (0.540 − 0.841i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.286 - 0.958i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (2963, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.286 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9362690018 - 1.256976330i\)
\(L(\frac12)\) \(\approx\) \(0.9362690018 - 1.256976330i\)
\(L(1)\) \(\approx\) \(1.047696431 - 0.2734771440i\)
\(L(1)\) \(\approx\) \(1.047696431 - 0.2734771440i\)

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

−18.01636607327743189251686797981, −17.532050315249279103265929540616, −16.61161216427382860752099899126, −16.171519087280739313104960479888, −15.31761745722736373541569346132, −14.84078778042177588582734073597, −14.25932253061322720170242489518, −13.32406579535297700443747142501, −12.61489589180366843965786438091, −12.268591762813113755158834921996, −11.3916721076938632441163144943, −10.81786618013913653204018823275, −9.93590025805648893037632513947, −9.24583169990560711500100611984, −8.79017352693764857488803297657, −7.974211797394410630584401214909, −7.21377907573960818522993331772, −6.34369826409556973303905320491, −5.872264792152854300820024579078, −5.119982277118343450365936948, −4.18688811382759119110900497415, −3.51179636524727871627002477583, −2.709414992460751806952275483970, −1.84385422719524578206096468345, −1.09884170427009856167676915982, 0.43749155664291207871819907593, 1.29187901089239640222147801015, 2.07709117918004743385934444480, 3.335899748177742580411893027940, 3.78208429681378841653871748846, 4.31697225336952531797004832646, 5.49244651602477796618160959423, 6.13814424236931428495559577299, 6.68979118691753728795942472947, 7.58346596687448572401280567869, 8.129085505012261405996376710279, 9.05204614038602890312984189159, 9.51740240391163721536590997153, 10.46401339744499991432713187822, 10.982775976616986403404622762697, 11.52211801763748376585629817739, 12.4797001491861079862169710072, 13.11508014820044927334497852661, 13.65909571880953276137904923144, 14.4919114777656175546781459335, 14.751821057092728910093463134343, 15.97306777838231601237753092571, 16.369744580674183173326554151650, 16.98001567398100351279515348092, 17.46302192417313031794835030120

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