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Properties

Label 1-5355-5355.1489-r0-0-0
Degree $1$
Conductor $5355$
Sign $-0.998 + 0.0617i$
Analytic cond. $24.8685$
Root an. cond. $24.8685$
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  + (0.707 + 0.707i)2-s + i·4-s + (−0.707 + 0.707i)8-s + (0.991 − 0.130i)11-s + (−0.866 + 0.5i)13-s − 16-s + (0.258 − 0.965i)19-s + (0.793 + 0.608i)22-s + (−0.608 − 0.793i)23-s + (−0.965 − 0.258i)26-s + (0.130 − 0.991i)29-s + (0.382 + 0.923i)31-s + (−0.707 − 0.707i)32-s + (−0.608 + 0.793i)37-s + (0.866 − 0.5i)38-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + (−0.707 + 0.707i)8-s + (0.991 − 0.130i)11-s + (−0.866 + 0.5i)13-s − 16-s + (0.258 − 0.965i)19-s + (0.793 + 0.608i)22-s + (−0.608 − 0.793i)23-s + (−0.965 − 0.258i)26-s + (0.130 − 0.991i)29-s + (0.382 + 0.923i)31-s + (−0.707 − 0.707i)32-s + (−0.608 + 0.793i)37-s + (0.866 − 0.5i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5355 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5355 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5355\)    =    \(3^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $-0.998 + 0.0617i$
Analytic conductor: \(24.8685\)
Root analytic conductor: \(24.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5355} (1489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5355,\ (0:\ ),\ -0.998 + 0.0617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05097813614 + 1.648854072i\)
\(L(\frac12)\) \(\approx\) \(0.05097813614 + 1.648854072i\)
\(L(1)\) \(\approx\) \(1.121862594 + 0.7399552015i\)
\(L(1)\) \(\approx\) \(1.121862594 + 0.7399552015i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.991 - 0.130i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (-0.608 - 0.793i)T \)
29 \( 1 + (0.130 - 0.991i)T \)
31 \( 1 + (0.382 + 0.923i)T \)
37 \( 1 + (-0.608 + 0.793i)T \)
41 \( 1 + (0.130 + 0.991i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.130 + 0.991i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (-0.965 + 0.258i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.130 - 0.991i)T \)
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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

−17.55947994937847227910906120725, −17.10903734582420238293467518474, −16.05062935442233911234267254874, −15.5475366632514443925543306365, −14.65040102363369979117012932412, −14.28998395107104370873116401410, −13.70833529286415266884888188669, −12.72936388887773928487590808844, −12.28931525687046596839372018655, −11.78680682824183736616607223119, −10.97071310130102696594275632741, −10.32339391941620246946587663379, −9.58806514488624536254611329148, −9.18183418726225557617531443448, −8.080632441271478371950210974092, −7.285891288416784620455733879545, −6.52340781978066857283707654577, −5.72216181525123101653612739671, −5.207800077873740000630238005910, −4.27803281821945159783200809718, −3.694397843411286844781358937541, −3.009888027868455160222609486935, −2.007916591020272565094361086598, −1.482892538683589474770818292583, −0.32233898732924847453406175442, 1.15791430289146289543871481171, 2.34992167858247379791968673218, 2.926024543946033056922283035102, 3.923296437779305635663307040323, 4.50029512339790419010901909472, 5.09607866625010698053714880932, 6.02906792991593621221345141954, 6.71955454632171805642827127215, 7.060382647905169547698397111662, 8.084847847192648593936350025966, 8.60120374973297747772188508889, 9.43312028913753287059786411718, 10.0372423063314204158523649013, 11.27353238842444858739993001114, 11.67306695012949804077433732334, 12.35177612088340753628758395678, 12.977263859283445444709716333593, 13.9068402326120117181076794278, 14.18228583963931258311486254530, 14.89766129227527768497539154112, 15.56499440334888476121550407563, 16.20081808447338134670403532250, 16.94828741908798640363884817826, 17.29933250213050953715338718602, 18.034874245561889634033791652565

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