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Properties

Label 1-6025-6025.38-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.999 + 0.0200i$
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  + (−0.544 − 0.838i)2-s + (0.358 + 0.933i)3-s + (−0.406 + 0.913i)4-s + (0.587 − 0.809i)6-s + (0.793 − 0.608i)7-s + (0.987 − 0.156i)8-s + (−0.743 + 0.669i)9-s + (0.566 − 0.824i)11-s + (−0.998 − 0.0523i)12-s + (0.824 − 0.566i)13-s + (−0.942 − 0.333i)14-s + (−0.669 − 0.743i)16-s + (−0.972 − 0.233i)17-s + (0.965 + 0.258i)18-s + (−0.999 + 0.0261i)19-s + ⋯
L(s)  = 1  + (−0.544 − 0.838i)2-s + (0.358 + 0.933i)3-s + (−0.406 + 0.913i)4-s + (0.587 − 0.809i)6-s + (0.793 − 0.608i)7-s + (0.987 − 0.156i)8-s + (−0.743 + 0.669i)9-s + (0.566 − 0.824i)11-s + (−0.998 − 0.0523i)12-s + (0.824 − 0.566i)13-s + (−0.942 − 0.333i)14-s + (−0.669 − 0.743i)16-s + (−0.972 − 0.233i)17-s + (0.965 + 0.258i)18-s + (−0.999 + 0.0261i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.999 + 0.0200i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.999 + 0.0200i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004046555747 - 0.4036484072i\)
\(L(\frac12)\) \(\approx\) \(0.004046555747 - 0.4036484072i\)
\(L(1)\) \(\approx\) \(0.7847504780 - 0.1850362228i\)
\(L(1)\) \(\approx\) \(0.7847504780 - 0.1850362228i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.544 - 0.838i)T \)
3 \( 1 + (0.358 + 0.933i)T \)
7 \( 1 + (0.793 - 0.608i)T \)
11 \( 1 + (0.566 - 0.824i)T \)
13 \( 1 + (0.824 - 0.566i)T \)
17 \( 1 + (-0.972 - 0.233i)T \)
19 \( 1 + (-0.999 + 0.0261i)T \)
23 \( 1 + (0.0784 - 0.996i)T \)
29 \( 1 + (-0.629 + 0.777i)T \)
31 \( 1 + (-0.725 - 0.688i)T \)
37 \( 1 + (0.333 + 0.942i)T \)
41 \( 1 + (-0.453 - 0.891i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + (0.156 - 0.987i)T \)
53 \( 1 + (0.358 + 0.933i)T \)
59 \( 1 + (-0.998 - 0.0523i)T \)
61 \( 1 + (-0.453 + 0.891i)T \)
67 \( 1 + (-0.933 - 0.358i)T \)
71 \( 1 + (0.878 - 0.477i)T \)
73 \( 1 + (0.0784 - 0.996i)T \)
79 \( 1 + (-0.156 + 0.987i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.182 - 0.983i)T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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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

−17.93866631931107935217044673821, −17.47573172351528794371421812069, −17.03873841489048685431091627872, −16.01371493345438912469786501477, −15.26019925271518150996427421018, −14.8732329206555333140468765862, −14.24190202315763221210666222000, −13.59471565992855380023303709106, −12.93057953988084351431727900230, −12.19572687660224296114252597044, −11.22700084207988686320321939870, −11.015830954760503444472185508719, −9.712018557645624837277671605683, −9.0271969163880216871966400307, −8.72200347532594367126825105054, −7.964830579641649694056476927286, −7.336271942096677314831360907437, −6.66757214815540127620977344834, −6.10747518896886692080494712559, −5.42409638506069060922528091580, −4.44461357625672476379523447505, −3.83048195167162392913261438379, −2.38894658035349060304966870106, −1.75491605830279700474070560959, −1.36129042725408602105553785586, 0.11365946614718814290643692171, 1.18608970077543769569300714335, 2.0423519668872871026842638140, 2.861518966333504590737594953157, 3.63828553441966149101656996256, 4.19032034976434544366955674481, 4.72005797035822479492200989314, 5.70577837660626967022301568693, 6.66834941804037525070867261158, 7.65638746828131457589823911735, 8.34766425437247286246330436163, 8.791032554138672710746317932608, 9.27146456177218691528965563173, 10.35032737390814578213285622327, 10.766821111086254464436631689245, 11.09958170359826989085746798362, 11.77833123213428015246012676028, 12.813847513930960979612350737279, 13.52002697227713067553594858128, 13.94616883307434330288903055789, 14.79570489349640927068704500732, 15.379745081242159617791547087338, 16.36479788817119812131423897081, 16.80991978835147154124368082554, 17.225476170367249706875317983550

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