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Properties

Label 1-5e3-125.44-r0-0-0
Degree $1$
Conductor $125$
Sign $0.837 + 0.546i$
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  + (0.187 − 0.982i)2-s + (−0.968 + 0.248i)3-s + (−0.929 − 0.368i)4-s + (0.0627 + 0.998i)6-s + (−0.309 + 0.951i)7-s + (−0.535 + 0.844i)8-s + (0.876 − 0.481i)9-s + (−0.187 + 0.982i)11-s + (0.992 + 0.125i)12-s + (−0.876 + 0.481i)13-s + (0.876 + 0.481i)14-s + (0.728 + 0.684i)16-s + (0.929 − 0.368i)17-s + (−0.309 − 0.951i)18-s + (0.968 + 0.248i)19-s + ⋯
L(s)  = 1  + (0.187 − 0.982i)2-s + (−0.968 + 0.248i)3-s + (−0.929 − 0.368i)4-s + (0.0627 + 0.998i)6-s + (−0.309 + 0.951i)7-s + (−0.535 + 0.844i)8-s + (0.876 − 0.481i)9-s + (−0.187 + 0.982i)11-s + (0.992 + 0.125i)12-s + (−0.876 + 0.481i)13-s + (0.876 + 0.481i)14-s + (0.728 + 0.684i)16-s + (0.929 − 0.368i)17-s + (−0.309 − 0.951i)18-s + (0.968 + 0.248i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.837 + 0.546i$
Analytic conductor: \(0.580497\)
Root analytic conductor: \(0.580497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (0:\ ),\ 0.837 + 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5707849048 + 0.1697245638i\)
\(L(\frac12)\) \(\approx\) \(0.5707849048 + 0.1697245638i\)
\(L(1)\) \(\approx\) \(0.6964298937 - 0.08081001513i\)
\(L(1)\) \(\approx\) \(0.6964298937 - 0.08081001513i\)

Euler product

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

−28.88900333321224516577642619886, −27.481971093945402593076551370919, −26.86399091483676617173666947265, −25.782346263847313371622164473021, −24.32554633037120615342982014157, −24.03294659884554466422130904787, −22.748713264952204522918128163, −22.30649468561101232361467224981, −20.983385091867023256510952954834, −19.26512695249948386052351330397, −18.31741363530831058309058587732, −17.05856776405919725776512524662, −16.70378100587279932386618217663, −15.59611550716207431411491645647, −14.18779156880425189268244446004, −13.21507913206546687084712044280, −12.24813366668283257353386443010, −10.75617505589986362305908429084, −9.66810571479367532975641838613, −7.910425780151890400222606671190, −7.093634389939693542104970292, −5.91319712391387894775541576305, −4.962410373846631182850933337656, −3.524356997872943610238344303753, −0.61505980075883672386345228347, 1.71527401902268717337811086307, 3.32705776739543969403339989692, 4.9037766665637476387441483927, 5.580515690224082403222644835120, 7.29421864497839778142856491585, 9.34951649464670177640639466636, 9.8614541573063405792455355234, 11.24847091108867540585011641194, 12.17475804107314073761210204716, 12.71401056204855701875457875855, 14.39639789957598587458187407380, 15.490494229354701009498920854879, 16.809119683974940380612930940047, 18.0028730532127612975713720450, 18.654186234024990628447311073370, 19.90681090585772059140556682943, 21.14189581526058439730070573296, 21.8895451478899005517053579669, 22.72062213986984035952424320438, 23.53164944277829635913689783896, 24.81058013609208802165680751743, 26.29445570645812875387004611315, 27.520580638247711976698164737386, 28.07572824784176887926558558182, 29.054138536829449205046459522507

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