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Properties

Label 1-5520-5520.4883-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.890 - 0.454i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.890 - 0.454i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (4883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.890 - 0.454i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246711159 - 0.2994642290i\)
\(L(\frac12)\) \(\approx\) \(1.246711159 - 0.2994642290i\)
\(L(1)\) \(\approx\) \(0.9612316370 + 0.03849961176i\)
\(L(1)\) \(\approx\) \(0.9612316370 + 0.03849961176i\)

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

−17.89796907491952946517639916033, −17.34512477337615863426618058955, −16.58840185170381252310981402580, −15.95994406237603297836480203044, −15.35745748871200628763940291544, −14.47461986036501771147565599720, −13.97995756795843866666763530324, −13.39472955482908295426519498815, −12.63774557679085377452548472882, −11.90290786136323587895723067933, −11.18333053835517448877543223921, −10.614038863985669970559615023496, −9.95717875044702022191202090925, −9.193620671346007897331826797474, −8.496514113931946274152570900745, −7.56999486308362152593829880710, −7.17527626656388568109760886049, −6.587008755506580767742653980622, −5.22421288814533691896429889916, −4.977234584823794795559148948111, −4.28528848280704242185102198111, −3.207484131205838782934716618915, −2.57233605360960765802965162453, −1.6789379751951135561825594395, −0.71588824420834816980834110420, 0.43329968399681110314143888821, 1.83078123103401612656514823744, 2.256403390530273064047453650239, 3.12559427432321153978476785122, 4.010322260026722945166755861608, 4.862746352956793688901270109313, 5.63279591347191985709031662580, 5.89244029964638889318236445097, 7.08935194143854485088446072420, 7.803945489283660895764681269809, 8.315224211426811558077596385990, 8.938560471561828683753114059157, 10.06708750878101334647260884568, 10.23083583363439182330586327246, 11.25979254294764096545804367270, 11.82324059416477940911456951948, 12.59540915774194825400966574522, 13.00778403837501700683528634056, 13.92397493018461119140723875652, 14.62860414742655925318784284543, 15.28587172471271902520983474435, 15.593494327575498066742045568327, 16.571026217080027957632338871377, 17.21477070291441006788197711736, 17.838104482802795247525468449846

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