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Properties

Label 1-5520-5520.1643-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.804 - 0.593i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.804 - 0.593i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (1643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.804 - 0.593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.190113193 - 0.3915361485i\)
\(L(\frac12)\) \(\approx\) \(1.190113193 - 0.3915361485i\)
\(L(1)\) \(\approx\) \(0.9431953299 + 0.02387160035i\)
\(L(1)\) \(\approx\) \(0.9431953299 + 0.02387160035i\)

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

−18.017763273342033101095093162860, −17.20291357411426894367924083162, −16.36861658720238435206238343856, −16.21331322387491394784868140093, −15.42903682105129523061763818288, −14.5370530275852176256531011668, −13.85172237551295636784420388500, −13.33017938224292701045286323619, −12.85288650636743063030575595130, −11.82824448670641447985424729154, −11.29834728131069033614641006315, −10.55935040078029312853182615916, −9.811720139984963061715981895240, −9.47664900310231043488744009890, −8.30871059950940744513988586678, −7.8259203951181673624122873746, −7.14861308748669356284092255949, −6.26145578792881793085609933931, −5.8009964460111308166408609809, −4.74611848389195323604426008125, −4.16292420539064662965105391519, −3.27328237792778399249962124446, −2.73090574075382722027906873882, −1.58468706471529936841739637783, −0.76904705000987329967120717586, 0.420681277781331670818983348413, 1.63065331555062791201652260448, 2.41059291425820635010002005368, 3.24474358942808306237010678216, 3.71551475366564157721312059068, 5.064789431624330329872630212929, 5.3325816055215066288174329437, 6.19968861129966558655566120221, 6.8544528897373043097097834195, 7.70984796517679707872219181246, 8.523383959264112179063816891681, 8.88661289889950830335712996780, 9.80757602271715692040718061794, 10.53103878433987516776000776214, 10.97810492915294962690847503952, 11.93477560033234990076392749094, 12.63486687946494370855806720130, 13.033050005358345174646584953046, 13.70926052173242058689465432223, 14.64720476808876873262266468082, 15.2808693848953289152697501824, 15.851948885351736267528613099104, 16.22619906780969271887604458863, 17.25371344070236327774998949955, 17.89396673148733969198200334550

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