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Properties

Label 1-1375-1375.1141-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.0443 + 0.999i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (−0.535 + 0.844i)2-s + (−0.425 − 0.904i)3-s + (−0.425 − 0.904i)4-s + (0.992 + 0.125i)6-s − 7-s + (0.992 + 0.125i)8-s + (−0.637 + 0.770i)9-s + (−0.637 + 0.770i)12-s + (0.637 − 0.770i)13-s + (0.535 − 0.844i)14-s + (−0.637 + 0.770i)16-s + (−0.728 − 0.684i)17-s + (−0.309 − 0.951i)18-s + (−0.728 − 0.684i)19-s + (0.425 + 0.904i)21-s + ⋯
L(s)  = 1  + (−0.535 + 0.844i)2-s + (−0.425 − 0.904i)3-s + (−0.425 − 0.904i)4-s + (0.992 + 0.125i)6-s − 7-s + (0.992 + 0.125i)8-s + (−0.637 + 0.770i)9-s + (−0.637 + 0.770i)12-s + (0.637 − 0.770i)13-s + (0.535 − 0.844i)14-s + (−0.637 + 0.770i)16-s + (−0.728 − 0.684i)17-s + (−0.309 − 0.951i)18-s + (−0.728 − 0.684i)19-s + (0.425 + 0.904i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0443 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0443 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.0443 + 0.999i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.0443 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02245411449 + 0.02347390241i\)
\(L(\frac12)\) \(\approx\) \(0.02245411449 + 0.02347390241i\)
\(L(1)\) \(\approx\) \(0.4848782650 - 0.05814888038i\)
\(L(1)\) \(\approx\) \(0.4848782650 - 0.05814888038i\)

Euler product

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

−20.4275291288125283807128185909, −19.848836084715021379451242189107, −18.90923996186763434642378135231, −18.38172304923156117270598774053, −17.33820328801142550446851540910, −16.63015744280119542398089613011, −16.27422848547508186751354212842, −15.304305954039638092275401493565, −14.35358606324738626639506038763, −13.28174521489343471869247818027, −12.62910685589622970655025623514, −11.80657799028712173746709945715, −11.01359817175476410348301699097, −10.34142456581883068175923266389, −9.795745853288372179417893101408, −8.76414969191553628536873151285, −8.52655342680820101391178286547, −6.87014183867283536772542130352, −6.3034453975486941488208414558, −5.06273402523638043276956326522, −4.017518753593058190835873254879, −3.63033667845931174616327798344, −2.5565198039719893807299998836, −1.39640233035755916159143932627, −0.0158475437256885612711238988, 0.498158738475841799567672636767, 1.71573402036539524319488009630, 2.797683025767417545311601074863, 4.13669524780321594551907939894, 5.3677725459528375137564711395, 5.95881287694315040865138826710, 6.70261406591083258107034398996, 7.35269616397550082944681201852, 8.14973200711783106626282619104, 9.06365263380631347382093211542, 9.75312431185060692605336796323, 10.880574869397802772810293514396, 11.36335876073792257369196348029, 12.75467551541065349600168426225, 13.22209627838178039735298060140, 13.80226092078347292528730588308, 14.95656933248533781216340652245, 15.73726171777008858170525138910, 16.29971877533068689153235303564, 17.26479374810852808545208578179, 17.68249444702182880785698445469, 18.514392856698793771819403576346, 19.15391378008141120986901290998, 19.725045952523437475420608580445, 20.53391104679948108534839380501

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