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Properties

Label 1-5e3-125.29-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  + (−0.728 − 0.684i)2-s + (−0.535 + 0.844i)3-s + (0.0627 + 0.998i)4-s + (0.968 − 0.248i)6-s + (−0.309 − 0.951i)7-s + (0.637 − 0.770i)8-s + (−0.425 − 0.904i)9-s + (0.728 + 0.684i)11-s + (−0.876 − 0.481i)12-s + (0.425 + 0.904i)13-s + (−0.425 + 0.904i)14-s + (−0.992 + 0.125i)16-s + (−0.0627 + 0.998i)17-s + (−0.309 + 0.951i)18-s + (0.535 + 0.844i)19-s + ⋯
L(s)  = 1  + (−0.728 − 0.684i)2-s + (−0.535 + 0.844i)3-s + (0.0627 + 0.998i)4-s + (0.968 − 0.248i)6-s + (−0.309 − 0.951i)7-s + (0.637 − 0.770i)8-s + (−0.425 − 0.904i)9-s + (0.728 + 0.684i)11-s + (−0.876 − 0.481i)12-s + (0.425 + 0.904i)13-s + (−0.425 + 0.904i)14-s + (−0.992 + 0.125i)16-s + (−0.0627 + 0.998i)17-s + (−0.309 + 0.951i)18-s + (0.535 + 0.844i)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: $\chi_{125} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (0:\ ),\ 0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5559638708 + 0.1962221626i\)
\(L(\frac12)\) \(\approx\) \(0.5559638708 + 0.1962221626i\)
\(L(1)\) \(\approx\) \(0.6375162533 + 0.05285279601i\)
\(L(1)\) \(\approx\) \(0.6375162533 + 0.05285279601i\)

Euler product

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

−28.610316491912323055365393702460, −27.8916740064057523211640154422, −26.85016071866387792619518129169, −25.4453249691101093809208044865, −24.85183898020020907921008442543, −24.13154610604755079579313746506, −22.84205145452240896938893048770, −22.16886034568168312846927342151, −20.259708991388017183844502171347, −19.17954536568985531359216468786, −18.42125557388917584063000982921, −17.651280766817532891230689150624, −16.49626090541424089540590697258, −15.64696847973035446894612689350, −14.300020953018819068242092990448, −13.159060223763061645887121539020, −11.79857904001653560718428527062, −10.856629201355077301986325852657, −9.28062069719167345347009258152, −8.34715674182345049931318334150, −7.08322536458150810957629501005, −6.08728301665785824730952429540, −5.217741183690566616235687336021, −2.60244541849935162433332608353, −0.845641054944707071146443106913, 1.44692117342387566464681345606, 3.6187841796113732292469829716, 4.26518644579186418333644685479, 6.27368747277038744158505572446, 7.57012534016382886606635168401, 9.2013859832797224088037801328, 9.87226017131253485175388924268, 10.9505344670048687469934425157, 11.786133122128076979272178426517, 13.045596164672067123470084704146, 14.526193227667968445231302343860, 16.00553460174953124157588526342, 16.88591630380668676856891354440, 17.47550658779747403557548246422, 18.86314897282056651295805731387, 20.0271060262145915116023853812, 20.7383861325239936888553696, 21.81067459440620780596157949359, 22.67833515389966864412904200606, 23.73362827512579333787870165436, 25.52575680130577429734665956604, 26.29648440754107413285931254769, 27.13801942228842803880673708477, 28.010391250212038340927771994949, 28.835188738070623228803175655380

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