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Properties

Label 1-4025-4025.284-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.980 - 0.198i$
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  + (−0.879 − 0.475i)2-s + (0.905 − 0.424i)3-s + (0.548 + 0.836i)4-s + (−0.998 − 0.0570i)6-s + (−0.0855 − 0.996i)8-s + (0.640 − 0.768i)9-s + (−0.432 + 0.901i)11-s + (0.851 + 0.524i)12-s + (0.736 + 0.676i)13-s + (−0.398 + 0.917i)16-s + (−0.449 − 0.893i)17-s + (−0.928 + 0.371i)18-s + (0.988 + 0.151i)19-s + (0.809 − 0.587i)22-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.879 − 0.475i)2-s + (0.905 − 0.424i)3-s + (0.548 + 0.836i)4-s + (−0.998 − 0.0570i)6-s + (−0.0855 − 0.996i)8-s + (0.640 − 0.768i)9-s + (−0.432 + 0.901i)11-s + (0.851 + 0.524i)12-s + (0.736 + 0.676i)13-s + (−0.398 + 0.917i)16-s + (−0.449 − 0.893i)17-s + (−0.928 + 0.371i)18-s + (0.988 + 0.151i)19-s + (0.809 − 0.587i)22-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.980 - 0.198i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.980 - 0.198i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.703158387 - 0.1707675046i\)
\(L(\frac12)\) \(\approx\) \(1.703158387 - 0.1707675046i\)
\(L(1)\) \(\approx\) \(1.028817866 - 0.2160008151i\)
\(L(1)\) \(\approx\) \(1.028817866 - 0.2160008151i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.879 - 0.475i)T \)
3 \( 1 + (0.905 - 0.424i)T \)
11 \( 1 + (-0.432 + 0.901i)T \)
13 \( 1 + (0.736 + 0.676i)T \)
17 \( 1 + (-0.449 - 0.893i)T \)
19 \( 1 + (0.988 + 0.151i)T \)
29 \( 1 + (-0.362 + 0.931i)T \)
31 \( 1 + (0.820 - 0.572i)T \)
37 \( 1 + (-0.640 + 0.768i)T \)
41 \( 1 + (0.897 + 0.441i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.104 + 0.994i)T \)
53 \( 1 + (-0.00951 - 0.999i)T \)
59 \( 1 + (-0.217 + 0.976i)T \)
61 \( 1 + (0.483 + 0.875i)T \)
67 \( 1 + (-0.380 + 0.924i)T \)
71 \( 1 + (0.941 - 0.336i)T \)
73 \( 1 + (-0.964 + 0.263i)T \)
79 \( 1 + (-0.948 - 0.318i)T \)
83 \( 1 + (0.466 - 0.884i)T \)
89 \( 1 + (0.797 - 0.603i)T \)
97 \( 1 + (0.466 + 0.884i)T \)
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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

−18.59113252403739576985580428219, −17.81775509282515693714579411265, −17.17720599686521557839604118954, −16.23792688337479135920980390101, −15.7501254235176685556006587743, −15.407167505650097480832686012000, −14.524195246813137438910655045558, −13.83508650193352551225068509480, −13.35413802057724154860624162745, −12.35520162550301069734037987308, −11.13578395132552230792937300611, −10.80626536948812732387906304215, −10.08057135606971023226194825714, −9.33278222885198428446898357342, −8.70843377078435115820320770135, −8.083847848668818078803324019085, −7.67834350688853168942233141713, −6.67494487854891332871996407116, −5.830198276982779430386420632318, −5.239028401251878685732203439205, −4.14125092340190192312313010201, −3.27140757383821069634325669780, −2.56538489025514644083564337869, −1.643543767301920350868052580013, −0.68021432467336754746413493642, 0.93534532952082960600440961039, 1.64359437182562565870747463565, 2.45830604813945597670592870976, 3.050758903928389344115751300739, 3.93509670489118741560064631329, 4.689954361250958331369948087088, 6.011875883176754238042380939, 7.01831263988463884498301172767, 7.27710325591776765525619094722, 8.08800774932349719646150166535, 8.82621418039138581262042880677, 9.380434450970745456141287426248, 9.93794213375658100877965229741, 10.762288237238401015021606017171, 11.69627449422180309899261978251, 12.10695122485004450909197788160, 13.05223877009351363398137439062, 13.478893029741759811739768582735, 14.30898002793344666700721638170, 15.14290602706070809662443476120, 15.96371160977784697356997058623, 16.21471274466885980723822838636, 17.535831423555362591036759905, 17.82956288907693973545215326783, 18.66382314368933959254247228767

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