↑

Properties

Label 1-6025-6025.5579-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.705 + 0.709i$
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  + (0.809 + 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + 6-s + (−0.809 − 0.587i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)11-s + (0.809 + 0.587i)12-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (0.809 − 0.587i)18-s + (−0.309 + 0.951i)19-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + 6-s + (−0.809 − 0.587i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)11-s + (0.809 + 0.587i)12-s + (−0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (0.809 − 0.587i)18-s + (−0.309 + 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.705 + 0.709i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (5579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.705 + 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.177893243 + 1.321489616i\)
\(L(\frac12)\) \(\approx\) \(3.177893243 + 1.321489616i\)
\(L(1)\) \(\approx\) \(1.886659100 + 0.4008441483i\)
\(L(1)\) \(\approx\) \(1.886659100 + 0.4008441483i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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

−17.67808371598557133401777805230, −16.67316473076070671135527185672, −16.07197917028867365575320037674, −15.51190128335225095329795959889, −14.91035383496205722240131964278, −14.35069357237536580298180379091, −13.59846578243301657311408013344, −13.23718261833777498285070531316, −12.43215922873859252851464447201, −11.63448515773670354666654482495, −11.17888578231585201723952047177, −10.32202554415465388482371834512, −9.56710083654014748375095000290, −9.10591360998598773687966075740, −8.73817100689281264168490330450, −7.261635832346821617516598412867, −6.86928200525759569287262231820, −5.96058276464548879809439840385, −5.143306695150186537192863049205, −4.575839860517357002615680938728, −3.69523928936357540262748047196, −3.28840296810077990355675269049, −2.35210567213783593692591091401, −2.05912152601633657368449565393, −0.664883706715360849514301066785, 0.84604807191575711682129408197, 2.0762463240603145849865683919, 2.590023516561140280404812000954, 3.46510087914317066599347648961, 4.11083999777268873651938156316, 4.57437207535775538120409778547, 5.86850908298907830863676635763, 6.45013555904528140717370768356, 6.88635658136538032379872089610, 7.68330108698094738709332509803, 8.12697811288313963263312303275, 9.0139504055205162357664930174, 9.688967528376873615924648315836, 10.414463715857827406073303816697, 11.47155971677805777862494049406, 12.30518969912737446577483611148, 12.684975894265952133853945215178, 13.211132068846741581647106369304, 13.87332456463919865378300240445, 14.55556545879589014974951262190, 15.0537771038929083637898687432, 15.49956434325170651364933454740, 16.54566924988089171190418265743, 17.07718686983618075142019749131, 17.51964585432545081513445861346

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