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Properties

Label 1-5520-5520.2123-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.308 + 0.951i$
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  + (0.540 + 0.841i)7-s + (0.989 + 0.142i)11-s + (0.841 + 0.540i)13-s + (−0.281 + 0.959i)17-s + (−0.281 − 0.959i)19-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (0.654 + 0.755i)37-s + (−0.654 + 0.755i)41-s + (−0.415 + 0.909i)43-s − i·47-s + (−0.415 + 0.909i)49-s + (−0.841 + 0.540i)53-s + (0.540 − 0.841i)59-s + (−0.909 + 0.415i)61-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)7-s + (0.989 + 0.142i)11-s + (0.841 + 0.540i)13-s + (−0.281 + 0.959i)17-s + (−0.281 − 0.959i)19-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (0.654 + 0.755i)37-s + (−0.654 + 0.755i)41-s + (−0.415 + 0.909i)43-s − i·47-s + (−0.415 + 0.909i)49-s + (−0.841 + 0.540i)53-s + (0.540 − 0.841i)59-s + (−0.909 + 0.415i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.308 + 0.951i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (2123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.308 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.048060974 + 1.442174816i\)
\(L(\frac12)\) \(\approx\) \(1.048060974 + 1.442174816i\)
\(L(1)\) \(\approx\) \(1.131685994 + 0.3139096122i\)
\(L(1)\) \(\approx\) \(1.131685994 + 0.3139096122i\)

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

−17.75108880900152765113799141854, −16.91788748262680178257954659606, −16.50727893219332399544535280736, −15.72294030037364169040781125694, −14.9882069232277004230657283263, −14.21242219718225060268205252878, −13.8503699549902322197436665253, −13.15596613639930941798615804694, −12.27503751866755002766473865571, −11.66057373379826274542039637613, −10.91965384886787768273800738845, −10.49602443532722557210980135862, −9.571512130189534345568480349771, −8.948389331827433010747571608529, −8.1405232935386088890598479317, −7.5785050238195885149087871056, −6.776124942710831947704208828114, −6.13153052956302698385965546306, −5.31942613363808682910096595113, −4.47969658882452442925859155890, −3.80005746498615701972059496975, −3.252709354720024092877290830452, −2.021614586099449325824364482561, −1.34867449192048500790619899141, −0.46122218246035074394466788377, 1.24117905128969074311977568193, 1.74971237600780631774038980095, 2.63617753905241882329348222471, 3.57659010770980501028578843012, 4.30238057641545853339616967522, 4.95490565954997815731496245021, 5.911933284896134699574210713304, 6.42039823269337756899051794062, 7.10001256318285605999072754404, 8.181751053391951973690083127658, 8.63630949901585954551244855949, 9.223339558950205906200596599044, 9.92409083576124588116253108765, 11.049415909806696087288380384195, 11.32100510358825471858103749189, 11.969295053885204924417294212899, 12.85833221040561707365885820062, 13.31437065355524168690832038778, 14.27349020934396742750951248082, 14.822640689525637612873689481189, 15.28910180689436001902910750017, 16.07532461101921102833690555805, 16.862262693105896981558619481816, 17.339250639757466418261333234295, 18.1712638204256107199601914001

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