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Properties

Label 1-6025-6025.629-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.0672 - 0.997i$
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  + (0.104 − 0.994i)2-s + (0.104 + 0.994i)3-s + (−0.978 − 0.207i)4-s + 6-s + (−0.104 + 0.994i)7-s + (−0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.104 − 0.994i)11-s + (0.104 − 0.994i)12-s + (0.913 − 0.406i)13-s + (0.978 + 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.809 + 0.587i)17-s + (0.104 + 0.994i)18-s + (0.978 − 0.207i)19-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (0.104 + 0.994i)3-s + (−0.978 − 0.207i)4-s + 6-s + (−0.104 + 0.994i)7-s + (−0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.104 − 0.994i)11-s + (0.104 − 0.994i)12-s + (0.913 − 0.406i)13-s + (0.978 + 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.809 + 0.587i)17-s + (0.104 + 0.994i)18-s + (0.978 − 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0672 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0672 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.0672 - 0.997i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.0672 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8090211773 - 0.8653935077i\)
\(L(\frac12)\) \(\approx\) \(0.8090211773 - 0.8653935077i\)
\(L(1)\) \(\approx\) \(0.9417770423 - 0.2027727989i\)
\(L(1)\) \(\approx\) \(0.9417770423 - 0.2027727989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T \)
3 \( 1 + (0.104 + 0.994i)T \)
7 \( 1 + (-0.104 + 0.994i)T \)
11 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (0.913 - 0.406i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.913 + 0.406i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
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

−17.83493152051547535701952257422, −17.327089879348100289909566348337, −16.57269937888168969270504364596, −16.01877818226246674621711959705, −15.20417629964074765870358948924, −14.5560342741713581563096785619, −13.807329314059962527041698487342, −13.40700196695162784885807976274, −13.048999661311652672951272629304, −12.02561143371165881986400039151, −11.50619708585542638828513699297, −10.56526886713860207686503201913, −9.54482083148491746184110305024, −9.19954438288459392847825815754, −8.23644883124565068359249818133, −7.625234071297301977194881114, −7.03719179527669153769412362198, −6.72848781758465089668986833569, −5.83219798789666756401284940572, −5.14878942826245665506087867840, −4.197002593471040394340281918327, −3.673153173713515910961843515782, −2.67937463572190560691961124624, −1.52758911615577688444699326517, −0.9107464881143509854698021370, 0.34292507070673350688966784095, 1.476686226311656256343506763817, 2.48505504692784617751170009632, 3.08604040884807241731968523690, 3.61908107694844281483107568516, 4.4120532113919743880387847685, 5.19819599256663356521174079163, 5.791086415932800770162577898996, 6.329439003938284257186347745443, 7.918533878788043649569052144794, 8.503341882499958802026169894280, 9.176138097658789255871261372191, 9.342135164410432704964478304127, 10.47957519918111477419047958225, 11.0407155929903613707898268746, 11.25442899027093104082728044164, 12.186618418447986302733679302912, 12.93030209194228670097113066225, 13.46446923839813651099010346890, 14.43345372111479058767163629236, 14.67816833864572836246994839184, 15.671865631778233680603237963057, 16.08151084304899813494537695277, 16.83238162924556790379343476216, 17.777442116402234881615667881106

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