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Properties

Label 1-5e3-125.4-r0-0-0
Degree $1$
Conductor $125$
Sign $0.778 + 0.627i$
Analytic cond. $0.580497$
Root an. cond. $0.580497$
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  + (0.992 + 0.125i)2-s + (0.637 + 0.770i)3-s + (0.968 + 0.248i)4-s + (0.535 + 0.844i)6-s + (−0.309 − 0.951i)7-s + (0.929 + 0.368i)8-s + (−0.187 + 0.982i)9-s + (−0.992 − 0.125i)11-s + (0.425 + 0.904i)12-s + (0.187 − 0.982i)13-s + (−0.187 − 0.982i)14-s + (0.876 + 0.481i)16-s + (−0.968 + 0.248i)17-s + (−0.309 + 0.951i)18-s + (−0.637 + 0.770i)19-s + ⋯
L(s)  = 1  + (0.992 + 0.125i)2-s + (0.637 + 0.770i)3-s + (0.968 + 0.248i)4-s + (0.535 + 0.844i)6-s + (−0.309 − 0.951i)7-s + (0.929 + 0.368i)8-s + (−0.187 + 0.982i)9-s + (−0.992 − 0.125i)11-s + (0.425 + 0.904i)12-s + (0.187 − 0.982i)13-s + (−0.187 − 0.982i)14-s + (0.876 + 0.481i)16-s + (−0.968 + 0.248i)17-s + (−0.309 + 0.951i)18-s + (−0.637 + 0.770i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(0.580497\)
Root analytic conductor: \(0.580497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (0:\ ),\ 0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.983136083 + 0.6999290268i\)
\(L(\frac12)\) \(\approx\) \(1.983136083 + 0.6999290268i\)
\(L(1)\) \(\approx\) \(1.891322361 + 0.4751750403i\)
\(L(1)\) \(\approx\) \(1.891322361 + 0.4751750403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.992 + 0.125i)T \)
3 \( 1 + (0.637 + 0.770i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.992 - 0.125i)T \)
13 \( 1 + (0.187 - 0.982i)T \)
17 \( 1 + (-0.968 + 0.248i)T \)
19 \( 1 + (-0.637 + 0.770i)T \)
23 \( 1 + (-0.728 - 0.684i)T \)
29 \( 1 + (0.0627 + 0.998i)T \)
31 \( 1 + (0.968 - 0.248i)T \)
37 \( 1 + (-0.876 - 0.481i)T \)
41 \( 1 + (0.728 - 0.684i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.929 - 0.368i)T \)
53 \( 1 + (-0.535 + 0.844i)T \)
59 \( 1 + (-0.425 - 0.904i)T \)
61 \( 1 + (0.728 + 0.684i)T \)
67 \( 1 + (-0.0627 + 0.998i)T \)
71 \( 1 + (-0.929 + 0.368i)T \)
73 \( 1 + (0.425 - 0.904i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (0.637 - 0.770i)T \)
89 \( 1 + (-0.425 + 0.904i)T \)
97 \( 1 + (-0.0627 - 0.998i)T \)
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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

−28.92234434098641175225305872517, −28.35302361452620114403055587447, −26.273442522256381835681420446551, −25.64013653664795746443580286963, −24.522146870824065005564925900862, −23.90491589969164880962654615170, −22.84810450815230268059607778351, −21.59565385386478574656166134704, −20.85744846915280705983017407886, −19.59981740022074967554375502096, −18.9142832589793534235665970410, −17.6806386419509547370483909975, −15.84115710057510013519185466542, −15.25923480508314519192729180249, −13.94144599123398937156821974764, −13.19711185493441884633327009338, −12.24164947782297198030687906341, −11.2642235527407068938588962573, −9.54621070649188747250729702734, −8.2295379374073637431511781275, −6.88704306687263695862569858384, −5.96295571286741160270568492524, −4.430133463886562081493617374560, −2.82626276642262802792851757780, −2.04508361601298360036406591261, 2.407434164854219010887564944457, 3.60924629525721316403735736119, 4.55851062260684078247751198338, 5.87217131096718116067040177837, 7.41599780568716073110681317058, 8.44398256760311648281646973851, 10.43194360862252926737919650379, 10.663127500825878193811137417798, 12.61205716004752991047024846760, 13.49922151382770527885979540868, 14.38255715617764323869337815541, 15.54444897347169888914888126927, 16.15286026399139241777276609031, 17.37730367272880278466742665656, 19.24048791162292709081787836990, 20.36825984307172143903058139524, 20.74923688034047963795874415097, 22.01188956412297162584443695246, 22.82959833492005598227429243260, 23.8512167940083674303883909635, 25.0143264194401572714658908902, 26.01485979885040995096888645724, 26.64687596797440134937099994334, 28.036512418023871611799044611795, 29.27189778061570560010576320416

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