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Properties

Label 1-6025-6025.2672-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.800 - 0.599i$
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  + (−0.156 − 0.987i)2-s + (−0.156 − 0.987i)3-s + (−0.951 + 0.309i)4-s + (−0.951 + 0.309i)6-s + (0.996 + 0.0784i)7-s + (0.453 + 0.891i)8-s + (−0.951 + 0.309i)9-s + (−0.522 − 0.852i)11-s + (0.453 + 0.891i)12-s + (0.996 − 0.0784i)13-s + (−0.0784 − 0.996i)14-s + (0.809 − 0.587i)16-s + (−0.382 − 0.923i)17-s + (0.453 + 0.891i)18-s + (0.760 + 0.649i)19-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (−0.156 − 0.987i)3-s + (−0.951 + 0.309i)4-s + (−0.951 + 0.309i)6-s + (0.996 + 0.0784i)7-s + (0.453 + 0.891i)8-s + (−0.951 + 0.309i)9-s + (−0.522 − 0.852i)11-s + (0.453 + 0.891i)12-s + (0.996 − 0.0784i)13-s + (−0.0784 − 0.996i)14-s + (0.809 − 0.587i)16-s + (−0.382 − 0.923i)17-s + (0.453 + 0.891i)18-s + (0.760 + 0.649i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.800 - 0.599i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2672, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.800 - 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5504526150 - 1.652896167i\)
\(L(\frac12)\) \(\approx\) \(0.5504526150 - 1.652896167i\)
\(L(1)\) \(\approx\) \(0.7021071676 - 0.7518813862i\)
\(L(1)\) \(\approx\) \(0.7021071676 - 0.7518813862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.156 - 0.987i)T \)
3 \( 1 + (-0.156 - 0.987i)T \)
7 \( 1 + (0.996 + 0.0784i)T \)
11 \( 1 + (-0.522 - 0.852i)T \)
13 \( 1 + (0.996 - 0.0784i)T \)
17 \( 1 + (-0.382 - 0.923i)T \)
19 \( 1 + (0.760 + 0.649i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (0.923 - 0.382i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.987 - 0.156i)T \)
43 \( 1 + (-0.0784 - 0.996i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.996 + 0.0784i)T \)
79 \( 1 + (0.453 + 0.891i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.972 + 0.233i)T \)
97 \( 1 + T \)
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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

−17.80380544840324710529012636040, −17.45481110777813332225529746553, −16.50569979237727343118110234850, −15.94749892930473777423063494964, −15.32049377719613897151725969287, −15.01694205059366258723515741558, −14.25014475290937296513492347104, −13.586758254559259003401715530941, −12.9899308723251014053586508293, −11.8559644352753763860270980306, −11.27085273370074102303232268777, −10.44155126288553562146156722017, −10.06006945820586243985281090898, −9.105978926609327395934497008840, −8.6763676118188120403926478865, −7.9480806110886673788231208096, −7.3310533933385063355860966603, −6.38489449512659876327387692536, −5.71201272094350985435083024823, −5.04963029023610722208843441124, −4.47839060445742633289599108984, −3.94243107155817987328833596855, −2.966194617293539552237965713128, −1.76365948273069568466652203811, −0.80982898459124333865037244332, 0.73922617334823888405399701852, 1.12667882766936681417221469711, 2.04898199791239397670352057715, 2.785748228325245786287449047960, 3.35819960037846165079858224460, 4.48929092537059712601397621613, 5.157723501221997708667658331856, 5.79214817783923073548032396171, 6.68529374896830975367113086173, 7.64596624361196321794419388760, 8.243793025415755223230116288454, 8.590761490967747456033365106171, 9.386543856083161357849883298446, 10.62291860566525875615217174290, 10.80338378334915531050846874771, 11.66872614771516243330382673100, 11.9400111067117580828791158082, 12.805857791732582317390165979878, 13.476492274512956531800029126285, 13.99508878071855219365294502989, 14.32741932899048731909623033236, 15.52219901322068862517810020826, 16.35751175543989479883015564368, 17.03629674373126030922657700117, 17.792618508414953197121312031532

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