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Properties

Label 1-1375-1375.1163-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.551 - 0.834i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (−0.998 + 0.0627i)2-s + (−0.125 − 0.992i)3-s + (0.992 − 0.125i)4-s + (0.187 + 0.982i)6-s + i·7-s + (−0.982 + 0.187i)8-s + (−0.968 + 0.248i)9-s + (−0.248 − 0.968i)12-s + (−0.248 − 0.968i)13-s + (−0.0627 − 0.998i)14-s + (0.968 − 0.248i)16-s + (−0.904 − 0.425i)17-s + (0.951 − 0.309i)18-s + (−0.425 + 0.904i)19-s + (0.992 − 0.125i)21-s + ⋯
L(s)  = 1  + (−0.998 + 0.0627i)2-s + (−0.125 − 0.992i)3-s + (0.992 − 0.125i)4-s + (0.187 + 0.982i)6-s + i·7-s + (−0.982 + 0.187i)8-s + (−0.968 + 0.248i)9-s + (−0.248 − 0.968i)12-s + (−0.248 − 0.968i)13-s + (−0.0627 − 0.998i)14-s + (0.968 − 0.248i)16-s + (−0.904 − 0.425i)17-s + (0.951 − 0.309i)18-s + (−0.425 + 0.904i)19-s + (0.992 − 0.125i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.551 - 0.834i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.551 - 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2678795624 - 0.4980895370i\)
\(L(\frac12)\) \(\approx\) \(0.2678795624 - 0.4980895370i\)
\(L(1)\) \(\approx\) \(0.5619676409 - 0.1841158351i\)
\(L(1)\) \(\approx\) \(0.5619676409 - 0.1841158351i\)

Euler product

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

−21.12543357254662792460073172840, −20.15142853893221654258137089404, −19.68654494211762173474358041046, −19.03231438499057282154827132185, −17.63534461765650901488046724270, −17.42834102873278289302179224281, −16.63147836899340465500634352380, −15.96756815170625197957130107465, −15.2870695676002661258612112256, −14.44952936170449571465638997538, −13.56982402458697471409047554355, −12.4259626086063451709350004411, −11.35525906963248474375817144940, −10.94780520098700406631523738898, −10.2688312296052279411210016797, −9.372343601528633281036723407, −8.90259335110992978965293885290, −7.94567861673980388387871242372, −6.86665671890285735420189806515, −6.41019663610358246590032132887, −5.0088524124439676261940012381, −4.24051089060402223741220349726, −3.29057988062228509210640487068, −2.29117882138675942357121343323, −1.0197915449976843180339337178, 0.3554069152270692654709490197, 1.52313687914391281523443008047, 2.50487146173355509557461414935, 2.967356870490986146696687348532, 4.83710718468050661319830543220, 5.90867322252566128596883815869, 6.38234979206644729067137767520, 7.29369845538490723299292218628, 8.30242499150566104154135617604, 8.48228707052196977570071555250, 9.611872595101119913611033474471, 10.462176155415420541859023475713, 11.43086781204770128769275995769, 11.977841377937965433175224671293, 12.72065406674206433159118388704, 13.52662104640940724270370710127, 14.8477742325006135414620464782, 15.17329292589742495906561806271, 16.24066368652275444465235305480, 17.03036322093752470093942494340, 17.75197056134825024510144067534, 18.3268549158699417098532204103, 18.90605183997432813580873567564, 19.58190389046367748482247112592, 20.33790889849637453140211228879

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