Properties

Label 1-6025-6025.1963-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: $\chi_{6025} (1963, \cdot )$
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

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

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