↑

Properties

Label 1-6025-6025.852-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.680 + 0.733i$
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

Related objects

Downloads

Learn more

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 + (0.878 − 0.477i)7-s + (−0.987 − 0.156i)8-s + (0.743 − 0.669i)9-s + (−0.902 − 0.430i)11-s + (−0.965 + 0.258i)12-s + (−0.991 + 0.130i)13-s + (−0.902 + 0.430i)14-s + (0.978 + 0.207i)16-s + (0.760 + 0.649i)17-s + (−0.777 + 0.629i)18-s + (0.878 − 0.477i)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 + (0.878 − 0.477i)7-s + (−0.987 − 0.156i)8-s + (0.743 − 0.669i)9-s + (−0.902 − 0.430i)11-s + (−0.965 + 0.258i)12-s + (−0.991 + 0.130i)13-s + (−0.902 + 0.430i)14-s + (0.978 + 0.207i)16-s + (0.760 + 0.649i)17-s + (−0.777 + 0.629i)18-s + (0.878 − 0.477i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.680 + 0.733i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (852, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.680 + 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07638883315 + 0.1750951570i\)
\(L(\frac12)\) \(\approx\) \(0.07638883315 + 0.1750951570i\)
\(L(1)\) \(\approx\) \(0.4873649853 + 0.001196871614i\)
\(L(1)\) \(\approx\) \(0.4873649853 + 0.001196871614i\)

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 \( 1 + (0.878 - 0.477i)T \)
11 \( 1 + (-0.902 - 0.430i)T \)
13 \( 1 + (-0.991 + 0.130i)T \)
17 \( 1 + (0.760 + 0.649i)T \)
19 \( 1 + (0.878 - 0.477i)T \)
23 \( 1 + (0.233 - 0.972i)T \)
29 \( 1 + (-0.838 + 0.544i)T \)
31 \( 1 + (-0.333 - 0.942i)T \)
37 \( 1 + (-0.878 + 0.477i)T \)
41 \( 1 + (0.156 + 0.987i)T \)
43 \( 1 + (-0.522 - 0.852i)T \)
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 \( 1 + (-0.824 - 0.566i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (-0.891 + 0.453i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (0.333 + 0.942i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.505772063951936885107125725128, −17.12044117712143977487187255584, −16.10665311148589917719957174060, −15.91347773376666075345567303526, −14.96538765482829914495298662267, −14.440689992810832144905353050743, −13.39917690141810957947754580319, −12.52425513533754119612454116335, −11.96775126513113146676279895200, −11.549950036660272608081405713137, −10.87028203733027477837840880884, −9.99540579755486855620078048138, −9.796963633961693500740738357997, −8.71690955085849420756976399111, −7.90979581504591774257982567076, −7.3217516020133106799745924921, −7.13044694898414585547954271967, −5.828639151629394345760036341334, −5.27993232572337630881220321352, −5.030751400483149503894085535821, −3.57237629464356019261462107668, −2.54328092074668100455080778660, −1.88426526470050167034800327788, −1.21143493043732014023407644843, −0.097082322423698621567389464402, 0.85609680452041341493890465752, 1.59909932596881458459939391789, 2.566799105965029853403760056698, 3.44915468554723888070575521361, 4.389904034426746543036479587360, 5.29836731149165209405708646936, 5.59523361255702243209581982095, 6.72840366017446057922299537410, 7.23561836919958924923902805454, 7.93209293116399455119681956772, 8.52537438747397925922432775674, 9.62030046960104578812965326362, 9.94224062573918985400688441755, 10.79578917584524670748948069495, 11.066458927358213146178537928467, 11.78967217406769739473694456895, 12.44450410422877116811363403137, 13.1191392399705421445408315552, 14.27144154331761713010745292019, 14.906319002751502839301658118485, 15.47801080176114800148436811136, 16.33786495907484137920734225777, 16.71739655888224254353833303505, 17.31674720794198496972287801423, 17.79113981878872168618707865886

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