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Properties

Label 1-5520-5520.1187-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.0706 - 0.997i$
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  + (−0.755 + 0.654i)7-s + (−0.540 + 0.841i)11-s + (−0.654 + 0.755i)13-s + (−0.909 + 0.415i)17-s + (−0.909 − 0.415i)19-s + (−0.909 + 0.415i)29-s + (0.142 + 0.989i)31-s + (0.959 + 0.281i)37-s + (−0.959 + 0.281i)41-s + (0.142 − 0.989i)43-s − i·47-s + (0.142 − 0.989i)49-s + (0.654 + 0.755i)53-s + (−0.755 − 0.654i)59-s + (−0.989 + 0.142i)61-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)7-s + (−0.540 + 0.841i)11-s + (−0.654 + 0.755i)13-s + (−0.909 + 0.415i)17-s + (−0.909 − 0.415i)19-s + (−0.909 + 0.415i)29-s + (0.142 + 0.989i)31-s + (0.959 + 0.281i)37-s + (−0.959 + 0.281i)41-s + (0.142 − 0.989i)43-s − i·47-s + (0.142 − 0.989i)49-s + (0.654 + 0.755i)53-s + (−0.755 − 0.654i)59-s + (−0.989 + 0.142i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0706 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0706 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.0706 - 0.997i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.0706 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02297661106 + 0.02466224518i\)
\(L(\frac12)\) \(\approx\) \(0.02297661106 + 0.02466224518i\)
\(L(1)\) \(\approx\) \(0.6736888686 + 0.1986307024i\)
\(L(1)\) \(\approx\) \(0.6736888686 + 0.1986307024i\)

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

−17.249684677455844974131682834329, −16.64991255332543097827853347365, −16.26804555188787851701617662615, −15.17668393694119478177869775466, −15.05830073646892714533940935921, −13.86509618953755773877754900144, −13.34850084358390850888806541166, −12.943937916010827500550076705615, −12.13526027925917696162774865794, −11.233811963015778533989182969475, −10.70046773682000645350375396496, −10.044599560505217520995871507447, −9.406202675932398371076166562258, −8.6017282222585968814114901904, −7.806673723961374045612565819991, −7.31197884048358416225147647700, −6.33284538627240405587174355212, −5.92087155893489335212562190674, −4.96145669752416290634235321405, −4.20718946202970199564334273359, −3.44191555867977860774735645213, −2.72520878542700421543016322376, −1.9734991743209804778124811006, −0.630128357843967932608700076148, −0.01231678638946286381659708960, 1.59803253708749391674378197997, 2.30402711863810377158522000530, 2.86776635640524219771255161527, 3.93784738041026561630137704573, 4.62544849333941349575383015600, 5.28044304762877576703512432736, 6.251893737600385115846587244567, 6.74991630518704426074284588092, 7.42293943008823205134014997380, 8.33867967122695737319275039459, 9.13462960777268592639435863417, 9.45405452611641268139780277319, 10.410514278156145984301606400798, 10.89025665303058452479644264323, 11.93695512434416682900542946808, 12.34570603309594930133762554867, 13.06672144007482282110801098982, 13.53108696991526372084560614324, 14.621090520946326180981921917554, 15.094504997180207169923518599829, 15.62754203851225780938520219422, 16.36966162625180690743694060708, 17.06065461260229226653764217442, 17.65669431519787260127501724411, 18.453543219776277027012341172433

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