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Properties

Label 1-5520-5520.4067-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.728 - 0.684i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (4067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.117672188 - 0.4425548130i\)
\(L(\frac12)\) \(\approx\) \(1.117672188 - 0.4425548130i\)
\(L(1)\) \(\approx\) \(0.9504234345 + 0.01096492241i\)
\(L(1)\) \(\approx\) \(0.9504234345 + 0.01096492241i\)

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

−17.73054967107085325061660325162, −17.25988011592742339889693464042, −16.81643625414751721070351136644, −16.080798179600829180506571987594, −15.040544126230090288246685056748, −14.7616273766219531517720057508, −14.052859431877990460633510458101, −13.29436711018324771074641360754, −12.69178613027486870007935758255, −11.96453744329826596285819153702, −11.25613372085916276699677917260, −10.559225390736717249935728603764, −10.040543276165146318091442099074, −9.0924974704031718688835887072, −8.7132349621059899873507123906, −7.59704823735221022105470890561, −7.02238757890737237529383735644, −6.696079822788541272529993517659, −5.52987512295524200207778256597, −4.80746774805192656084739759176, −4.03095319900101922200780296837, −3.7040500553129117671128767360, −2.17988333932821151188665142811, −1.981827920757741581269826973868, −0.735604715518791926477305723355, 0.38939078454157712361189537783, 1.74427106386585693188753673705, 2.2440218744840335008901944756, 3.12641061111180478697848069941, 3.96135210460675766178733309252, 4.812273166419835409683761468528, 5.40011598450744691755078956521, 6.31676343198299928237248657950, 6.69068567024184824328116497300, 7.84676260702593304190477687892, 8.29356583669446620358902980295, 9.209525194425948856066976276426, 9.46444768110644339200166894880, 10.51933147413791795845734752400, 11.36699979399933046513604380675, 11.58774770487417884002239143341, 12.635877698553836311587232237445, 12.93755605207014827395718804544, 13.926119105520542522189116043095, 14.63625870266150278816777565119, 15.089601068433783071065690120554, 15.66943958799953827794490704682, 16.6798266050881509117663359340, 17.005629875529337335472146169203, 17.8267687702572243649105281259

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