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Properties

Label 1-4025-4025.321-r0-0-0
Degree $1$
Conductor $4025$
Sign $-0.637 - 0.770i$
Analytic cond. $18.6920$
Root an. cond. $18.6920$
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  + (−0.809 − 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)6-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + 18-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)22-s − 24-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)6-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + 18-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)22-s − 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.637 - 0.770i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ -0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4575424154 - 0.9723271265i\)
\(L(\frac12)\) \(\approx\) \(0.4575424154 - 0.9723271265i\)
\(L(1)\) \(\approx\) \(0.6155352740 - 0.4148812918i\)
\(L(1)\) \(\approx\) \(0.6155352740 - 0.4148812918i\)

Euler product

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

−18.55204786554103492895832761375, −18.00339151303486244025208653220, −17.00648085945168481141203689914, −16.708795594365176698750594239672, −16.27156542744391540388572982402, −15.35241190436783168024621314138, −14.92130527651490659542541192212, −14.16798022779211200898798756189, −13.60862098416027562145947667833, −12.304450098463578431226120237415, −11.46308326858353013926794059525, −11.10352728459483914288386047693, −10.28847046065539300237007187319, −9.6093173925141347515335380977, −9.12878709344445638273188023342, −8.28902416694428112691003820946, −7.83387357504119190910371138691, −6.478855530107398979227768643967, −6.14281397636543725637843034948, −5.59193577141875871555228524251, −4.44247814590719918650648729876, −3.916334282146090278399267295581, −2.94288730440731822469035378440, −1.67483716369976413272030818301, −0.907713417179880532558385301327, 0.55856903753975642276205086357, 1.24508870236972685306004461015, 2.0245556132471543310948358924, 2.91190602506656472923445116754, 3.56972539234999500377608885511, 4.73527943619409141896052639458, 5.56826875526193295991287620585, 6.71814256126707678752063638094, 6.96706464897676368217941695440, 7.7657535721107969078257904229, 8.56232126424789222509962781185, 9.11167399470043416748100151508, 9.93224604131208874321489914391, 10.80234743028569435469311413768, 11.387712238716166925104114262722, 11.925296692776697061042126644, 12.664322294310134287109769365862, 13.16173126971717806337096655698, 13.955051755963537025139754824250, 14.72812419115824735116799283025, 15.90372605086855577745216793159, 16.29124267518587468310226251761, 17.19104360980843246268689735415, 17.79483605597752897744222201139, 18.12976188463098732270742989050

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