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Properties

Label 1-4025-4025.171-r0-0-0
Degree $1$
Conductor $4025$
Sign $-0.873 - 0.486i$
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  + (0.272 + 0.962i)2-s + (−0.953 − 0.299i)3-s + (−0.851 + 0.524i)4-s + (0.0285 − 0.999i)6-s + (−0.736 − 0.676i)8-s + (0.820 + 0.572i)9-s + (−0.999 + 0.0380i)11-s + (0.969 − 0.244i)12-s + (0.362 − 0.931i)13-s + (0.449 − 0.893i)16-s + (0.879 − 0.475i)17-s + (−0.327 + 0.945i)18-s + (−0.432 − 0.901i)19-s + (−0.309 − 0.951i)22-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.272 + 0.962i)2-s + (−0.953 − 0.299i)3-s + (−0.851 + 0.524i)4-s + (0.0285 − 0.999i)6-s + (−0.736 − 0.676i)8-s + (0.820 + 0.572i)9-s + (−0.999 + 0.0380i)11-s + (0.969 − 0.244i)12-s + (0.362 − 0.931i)13-s + (0.449 − 0.893i)16-s + (0.879 − 0.475i)17-s + (−0.327 + 0.945i)18-s + (−0.432 − 0.901i)19-s + (−0.309 − 0.951i)22-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.873 - 0.486i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ -0.873 - 0.486i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03252145067 - 0.1253239784i\)
\(L(\frac12)\) \(\approx\) \(0.03252145067 - 0.1253239784i\)
\(L(1)\) \(\approx\) \(0.6469156847 + 0.1589340139i\)
\(L(1)\) \(\approx\) \(0.6469156847 + 0.1589340139i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.272 + 0.962i)T \)
3 \( 1 + (-0.953 - 0.299i)T \)
11 \( 1 + (-0.999 + 0.0380i)T \)
13 \( 1 + (0.362 - 0.931i)T \)
17 \( 1 + (0.879 - 0.475i)T \)
19 \( 1 + (-0.432 - 0.901i)T \)
29 \( 1 + (-0.564 - 0.825i)T \)
31 \( 1 + (0.217 - 0.976i)T \)
37 \( 1 + (-0.820 - 0.572i)T \)
41 \( 1 + (-0.974 - 0.226i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (-0.964 + 0.263i)T \)
59 \( 1 + (-0.988 + 0.151i)T \)
61 \( 1 + (0.00951 - 0.999i)T \)
67 \( 1 + (0.0665 + 0.997i)T \)
71 \( 1 + (-0.985 + 0.170i)T \)
73 \( 1 + (-0.380 + 0.924i)T \)
79 \( 1 + (0.935 - 0.353i)T \)
83 \( 1 + (0.516 + 0.856i)T \)
89 \( 1 + (0.749 + 0.662i)T \)
97 \( 1 + (0.516 - 0.856i)T \)
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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

−18.7507434245293397169702912121, −18.264580247847217310942207987469, −17.51763620857947779972692742658, −16.70827140248811345834231600200, −16.21218529560232135422900079459, −15.27513897031680537760372977278, −14.64525550151793533227598000860, −13.81528256072118499573853529485, −13.10688469145066839677127334120, −12.3274170035335491342134635612, −12.0204497525285194840413459894, −11.13644891923504737170215583066, −10.493296385660351306212321980069, −10.1806411116933076437851840141, −9.28971150618347030911660812050, −8.52380578581087354875430893235, −7.62170504796750899340032666499, −6.51239271580591873261078206139, −5.903838805222321991449103113330, −5.11374344821741761262085035103, −4.622450832641734098480956508260, −3.66220464546913951833520575740, −3.14758199846969601312053627561, −1.786245448060212359323145522786, −1.35721464960882799953836886963, 0.0499877532628866890553059222, 0.83402840388958668405442619968, 2.24551670805541032487414814884, 3.22370147413279523928358705956, 4.16290469095211554848235554328, 5.040444385995966218904492107012, 5.481084682374878991669330100051, 6.08376307856858986230142723061, 6.931209471062923268762319130289, 7.62639004717959000375772099504, 8.04471997352670878448744287762, 9.04090299050626799373094006788, 9.968245722653947312060356853209, 10.54977046958422510633392942504, 11.40379852553314895822768471372, 12.22935598701821158565292690037, 12.81923515022651623254356911119, 13.41209102213543783865562209796, 13.9320390459615024635512002589, 15.160440372546214156825698421671, 15.48659866939643663093003723068, 16.097125997151558553118493751154, 16.91803281682378360654621984297, 17.384070673781289177961260053853, 17.97308365966568270774996956425

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