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Properties

Label 1-5e3-125.6-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  + (0.876 + 0.481i)2-s + (−0.929 − 0.368i)3-s + (0.535 + 0.844i)4-s + (−0.637 − 0.770i)6-s + (0.309 − 0.951i)7-s + (0.0627 + 0.998i)8-s + (0.728 + 0.684i)9-s + (0.876 + 0.481i)11-s + (−0.187 − 0.982i)12-s + (0.728 + 0.684i)13-s + (0.728 − 0.684i)14-s + (−0.425 + 0.904i)16-s + (0.535 − 0.844i)17-s + (0.309 + 0.951i)18-s + (−0.929 + 0.368i)19-s + ⋯
L(s)  = 1  + (0.876 + 0.481i)2-s + (−0.929 − 0.368i)3-s + (0.535 + 0.844i)4-s + (−0.637 − 0.770i)6-s + (0.309 − 0.951i)7-s + (0.0627 + 0.998i)8-s + (0.728 + 0.684i)9-s + (0.876 + 0.481i)11-s + (−0.187 − 0.982i)12-s + (0.728 + 0.684i)13-s + (0.728 − 0.684i)14-s + (−0.425 + 0.904i)16-s + (0.535 − 0.844i)17-s + (0.309 + 0.951i)18-s + (−0.929 + 0.368i)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: $\chi_{125} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (0:\ ),\ 0.837 + 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.355388338 + 0.4030286939i\)
\(L(\frac12)\) \(\approx\) \(1.355388338 + 0.4030286939i\)
\(L(1)\) \(\approx\) \(1.334132363 + 0.2807087821i\)
\(L(1)\) \(\approx\) \(1.334132363 + 0.2807087821i\)

Euler product

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

−28.72302197936104859578357204437, −28.01614384993660123744684747230, −27.36485332871138705632481445907, −25.53393829635626355464605450464, −24.472622104260034161894118672597, −23.54991575337041319227157058449, −22.63133414368077113975653770136, −21.64762923638250896409284694603, −21.24968484703564281444808392583, −19.773212347138048502766057800341, −18.65574631813239540029484811856, −17.528947285669849805687016550467, −16.14958769089330042417644949383, −15.31049038849100842948319545768, −14.286353371716074555039096270365, −12.762760531106655022837312663392, −12.00409557485438383087124073742, −11.0595506436810017821736921949, −10.113586107106744455578697060574, −8.59986273124312843528032699563, −6.42733658464210222022403041874, −5.805914270059288127276907540266, −4.585310703902825527403816702621, −3.34864554209826620602250904412, −1.48653377445096196280257650243, 1.69031920352050732810163173851, 3.90534772864894958269375224538, 4.7853749407050948414017762172, 6.25666708695835526252371043002, 6.95569418853528711071676994625, 8.15587685489877016389712953070, 10.18021186045311148650497665599, 11.48657503019383387915267625723, 12.1226354140341454135905522795, 13.485850119146193375402886995760, 14.17353251570338713301535057722, 15.63278951620467392398126121802, 16.77845326566182437753489752665, 17.224633850592737763313211323528, 18.55858830476600198606391969800, 20.081415871881983035381899807915, 21.14337819557416405788318471878, 22.21231394978545999397705634652, 23.19843276362566751344760190444, 23.63666382081397455986095575155, 24.73469255931536598012314026020, 25.69716549385625475782747969810, 26.99715316390689686565518383305, 28.03564277826804690822071568871, 29.36779777319458923614451267291

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