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Properties

Label 1-4025-4025.44-r1-0-0
Degree $1$
Conductor $4025$
Sign $0.394 + 0.918i$
Analytic cond. $432.546$
Root an. cond. $432.546$
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  + (−0.00951 − 0.999i)2-s + (0.851 + 0.524i)3-s + (−0.999 + 0.0190i)4-s + (0.516 − 0.856i)6-s + (0.0285 + 0.999i)8-s + (0.449 + 0.893i)9-s + (0.948 + 0.318i)11-s + (−0.861 − 0.508i)12-s + (−0.696 + 0.717i)13-s + (0.999 − 0.0380i)16-s + (0.483 + 0.875i)17-s + (0.888 − 0.458i)18-s + (−0.123 − 0.992i)19-s + (0.309 − 0.951i)22-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.00951 − 0.999i)2-s + (0.851 + 0.524i)3-s + (−0.999 + 0.0190i)4-s + (0.516 − 0.856i)6-s + (0.0285 + 0.999i)8-s + (0.449 + 0.893i)9-s + (0.948 + 0.318i)11-s + (−0.861 − 0.508i)12-s + (−0.696 + 0.717i)13-s + (0.999 − 0.0380i)16-s + (0.483 + 0.875i)17-s + (0.888 − 0.458i)18-s + (−0.123 − 0.992i)19-s + (0.309 − 0.951i)22-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.394 + 0.918i$
Analytic conductor: \(432.546\)
Root analytic conductor: \(432.546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (1:\ ),\ 0.394 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.280002823 + 1.501914738i\)
\(L(\frac12)\) \(\approx\) \(2.280002823 + 1.501914738i\)
\(L(1)\) \(\approx\) \(1.332994361 - 0.09756100792i\)
\(L(1)\) \(\approx\) \(1.332994361 - 0.09756100792i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.00951 - 0.999i)T \)
3 \( 1 + (0.851 + 0.524i)T \)
11 \( 1 + (0.948 + 0.318i)T \)
13 \( 1 + (-0.696 + 0.717i)T \)
17 \( 1 + (0.483 + 0.875i)T \)
19 \( 1 + (-0.123 - 0.992i)T \)
29 \( 1 + (-0.921 + 0.389i)T \)
31 \( 1 + (0.879 + 0.475i)T \)
37 \( 1 + (0.449 + 0.893i)T \)
41 \( 1 + (-0.362 - 0.931i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (0.978 - 0.207i)T \)
53 \( 1 + (0.640 - 0.768i)T \)
59 \( 1 + (0.272 + 0.962i)T \)
61 \( 1 + (0.761 + 0.647i)T \)
67 \( 1 + (-0.217 - 0.976i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (-0.820 - 0.572i)T \)
79 \( 1 + (0.0665 + 0.997i)T \)
83 \( 1 + (0.774 + 0.633i)T \)
89 \( 1 + (0.991 + 0.132i)T \)
97 \( 1 + (0.774 - 0.633i)T \)
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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

−18.19107342283437445353126412249, −17.409732359230454465046840246984, −16.84919146945050072082249888958, −16.1210534762124077313831909649, −15.27157307228955965621757973349, −14.72069622153649313093815372024, −14.24466221292673323104880134318, −13.60681565660205817099561599631, −12.905596479242383968667407771001, −12.23880874399791518881909985901, −11.53857342825633007687893814892, −10.112399797765096643269554490337, −9.70163586074973987168061391128, −8.97318778928738734130230457122, −8.26459910756709988663579198371, −7.63282260746782312217536346638, −7.14272235096251861186686128263, −6.25475463455897159492972438831, −5.69032371316139193751884255244, −4.68564432645030662503522448683, −3.828129365923994243808572841419, −3.23649808530486480205178333373, −2.20066027089528675008026645845, −1.09440736651958084173897494580, −0.40439702964477615032911150588, 0.99466510564601310263403657648, 1.867995922122242365654064267463, 2.446700254263497484187795640494, 3.39908237740734428008636097070, 3.96245720683646497296228737291, 4.65187253235803114735026436641, 5.300654039033420156184162481044, 6.56454174169174969371936971462, 7.39178024583298359850021169114, 8.30462528902359058910591109288, 8.93645617698775370886826991368, 9.43690839647834308667725628481, 10.124728793175311116334638101619, 10.69335694317468267619076060913, 11.64927317907338662472946888849, 12.10780204720699007976834215565, 13.04124664114792238087285597336, 13.58937947805130810153164989670, 14.355644639161871577501457836429, 14.823516499349916108677957036623, 15.426283839952964546028468162516, 16.70989906177660453569578825675, 16.959826206721072922529591229380, 17.8581408660002266180935097130, 18.78369651416111472610163034842

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