Properties

Label 1-6025-6025.3677-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.0828 - 0.996i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (0.998 + 0.0523i)2-s + (0.933 − 0.358i)3-s + (0.994 + 0.104i)4-s + (0.951 − 0.309i)6-s + (−0.477 − 0.878i)7-s + (0.987 + 0.156i)8-s + (0.743 − 0.669i)9-s + (−0.430 + 0.902i)11-s + (0.965 − 0.258i)12-s + (−0.130 − 0.991i)13-s + (−0.430 − 0.902i)14-s + (0.978 + 0.207i)16-s + (−0.649 + 0.760i)17-s + (0.777 − 0.629i)18-s + (−0.477 − 0.878i)19-s + ⋯
L(s)  = 1  + (0.998 + 0.0523i)2-s + (0.933 − 0.358i)3-s + (0.994 + 0.104i)4-s + (0.951 − 0.309i)6-s + (−0.477 − 0.878i)7-s + (0.987 + 0.156i)8-s + (0.743 − 0.669i)9-s + (−0.430 + 0.902i)11-s + (0.965 − 0.258i)12-s + (−0.130 − 0.991i)13-s + (−0.430 − 0.902i)14-s + (0.978 + 0.207i)16-s + (−0.649 + 0.760i)17-s + (0.777 − 0.629i)18-s + (−0.477 − 0.878i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0828 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0828 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.0828 - 0.996i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.0828 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.085259197 - 3.352421739i\)
\(L(\frac12)\) \(\approx\) \(3.085259197 - 3.352421739i\)
\(L(1)\) \(\approx\) \(2.338921559 - 0.7492197297i\)
\(L(1)\) \(\approx\) \(2.338921559 - 0.7492197297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.998 + 0.0523i)T \)
3 \( 1 + (0.933 - 0.358i)T \)
7 \( 1 + (-0.477 - 0.878i)T \)
11 \( 1 + (-0.430 + 0.902i)T \)
13 \( 1 + (-0.130 - 0.991i)T \)
17 \( 1 + (-0.649 + 0.760i)T \)
19 \( 1 + (-0.477 - 0.878i)T \)
23 \( 1 + (-0.972 - 0.233i)T \)
29 \( 1 + (0.838 - 0.544i)T \)
31 \( 1 + (0.942 - 0.333i)T \)
37 \( 1 + (0.477 + 0.878i)T \)
41 \( 1 + (-0.156 - 0.987i)T \)
43 \( 1 + (0.852 - 0.522i)T \)
47 \( 1 + (0.453 - 0.891i)T \)
53 \( 1 + (-0.0523 - 0.998i)T \)
59 \( 1 + (-0.629 + 0.777i)T \)
61 \( 1 + (-0.987 + 0.156i)T \)
67 \( 1 + (-0.998 - 0.0523i)T \)
71 \( 1 + (0.566 - 0.824i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (0.891 - 0.453i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.942 + 0.333i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−18.143336121371971692433592756611, −16.82421632743141351155003781760, −16.135289743680540982753997363000, −15.87167098401859360543178431608, −15.285911286364181890838763662220, −14.37244732886554911287880387017, −14.02710202943846442399561232512, −13.51538261157386454114462697885, −12.61239556959660235477674210683, −12.22764453970763438478223138099, −11.31217522292686869594041392764, −10.742308059831090305626738757902, −9.843594125701209313246568640012, −9.3022122446365546393526368471, −8.45307099704242522408748771613, −7.91017261157346541394928580702, −6.996290807793667400792211115034, −6.23133135898857135640190045356, −5.71322213073458954806697047959, −4.6701250131525398463254318458, −4.277724370032541255823816504666, −3.3362716242921632882082240871, −2.75641308119991234553788850947, −2.25594742273796141397197221803, −1.35388762299462419785845130791, 0.59801430691604212621710171527, 1.71754725904839127965152455818, 2.48350475798735817094785146643, 2.94038075486968186790982893173, 3.990160697145783258371841162341, 4.26014691792563101683855986909, 5.1197089115616913639008997101, 6.31820438770612373437816401312, 6.58683186991621788880959447287, 7.5008805231925470203893726371, 7.87291316907323995052364202083, 8.65852693216298499189795310490, 9.76563179086706794142109883903, 10.33338078627640369895396675630, 10.766985204832455232656952700353, 12.02317792935427984269605134566, 12.45199739789213368553746946999, 13.15577952578459686816380360625, 13.544757581121429674094814162491, 14.0484789487899987179225034714, 15.09677565257445025599448217974, 15.285359594398291443544331914765, 15.82350219738943652401283564381, 16.86975302871714278274349717526, 17.49561673966531945653612932056

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