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Properties

Label 1-5520-5520.4667-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.971 - 0.235i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.971 - 0.235i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (4667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06319997423 - 0.5282886003i\)
\(L(\frac12)\) \(\approx\) \(0.06319997423 - 0.5282886003i\)
\(L(1)\) \(\approx\) \(0.7934719818 - 0.1448607247i\)
\(L(1)\) \(\approx\) \(0.7934719818 - 0.1448607247i\)

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

−18.38576180704304128781591593996, −17.35001823191641312579602523311, −16.7666872081222978117838068890, −16.33385303006023362289239727059, −15.49050520070057233481979310321, −14.99859771938045960965978995904, −14.190218636358355591041177491633, −13.40545490934447355308475639297, −12.9490803017898057112378862890, −12.26144776614752688221917736894, −11.61747001498942705005087856018, −10.69299371124701898987100044754, −10.05065252694163198327235306875, −9.66991160248418168590389961688, −8.73195118695068398292394494545, −8.043808695610132150520058186822, −7.33148315409293974201876348594, −6.5486137442550364720389900045, −5.984273572355640057114388675441, −5.23999482003685702915531959707, −4.14857908403993068752671391944, −3.82625040223337599859291470620, −2.73518728422583898399487699452, −2.14185987520435462739001818102, −1.11167698580739108581979099382, 0.16282089706851273182974223141, 0.983462851409661655200767100701, 2.30787049422405167145552641376, 2.98921259014788887029548716354, 3.41525782624524426643938469191, 4.53578180580790641196028636319, 5.30299402000996825428999750181, 5.92303153265734211142798867566, 6.60236618588109597318040801961, 7.46650791984909618841563610968, 8.06149882602528245142996924637, 8.82160868681501103633703481479, 9.5998920153864275566003848542, 10.174151535794749375466146360635, 10.82900782755345464059662601090, 11.52110464662894341429727348134, 12.43656291641668876428733415221, 13.0894191334934911252805368029, 13.25868211589532059947082833040, 14.3236367269290665756416623916, 14.98454471598046975474242584968, 15.72594191716058179927326581761, 16.18133906390219509858867584994, 16.73822808135390002795605643475, 17.74206165155625210788942437884

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