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Properties

Label 1-1375-1375.1016-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.307 - 0.951i$
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  + (0.0627 + 0.998i)2-s + (0.728 − 0.684i)3-s + (−0.992 + 0.125i)4-s + (0.728 + 0.684i)6-s + (0.309 − 0.951i)7-s + (−0.187 − 0.982i)8-s + (0.0627 − 0.998i)9-s + (−0.637 + 0.770i)12-s + (0.0627 − 0.998i)13-s + (0.968 + 0.248i)14-s + (0.968 − 0.248i)16-s + (−0.187 − 0.982i)17-s + 18-s + (−0.425 + 0.904i)19-s + (−0.425 − 0.904i)21-s + ⋯
L(s)  = 1  + (0.0627 + 0.998i)2-s + (0.728 − 0.684i)3-s + (−0.992 + 0.125i)4-s + (0.728 + 0.684i)6-s + (0.309 − 0.951i)7-s + (−0.187 − 0.982i)8-s + (0.0627 − 0.998i)9-s + (−0.637 + 0.770i)12-s + (0.0627 − 0.998i)13-s + (0.968 + 0.248i)14-s + (0.968 − 0.248i)16-s + (−0.187 − 0.982i)17-s + 18-s + (−0.425 + 0.904i)19-s + (−0.425 − 0.904i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.307 - 0.951i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1016, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.307 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.257717270 - 0.9150912988i\)
\(L(\frac12)\) \(\approx\) \(1.257717270 - 0.9150912988i\)
\(L(1)\) \(\approx\) \(1.200123750 - 0.06436078714i\)
\(L(1)\) \(\approx\) \(1.200123750 - 0.06436078714i\)

Euler product

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

−21.06070248175897053342918695628, −20.37067793938405856719036441600, −19.34535377480447322316247057553, −19.15365083293961798530533853796, −18.21722286055979414140524852285, −17.33582757180364839551066954131, −16.44039921829240968044329845647, −15.39645211052634746326215036060, −14.817800496686916548562357192080, −14.115563370387039633618832249933, −13.327906744724658034540820522930, −12.49411570511818168091976139510, −11.66806535672205966955032920011, −10.89941312834021205193133998544, −10.186997987887417198264092241360, −9.31196038092007252076413939755, −8.59811798879595413968258476788, −8.31892277292659810688885652999, −6.7559029150390077486858988674, −5.58122361565375754953952511575, −4.6236091826207549614325610601, −4.16519056469036363527500946107, −2.99193649455496813466919720023, −2.35709387507344908527835001254, −1.53933262635869407077656176647, 0.53682272067544736818968629906, 1.574365854030232422361383585, 3.05126685638868763248898955570, 3.76311940234066963571860345265, 4.76792881040194873404703440731, 5.73063981196042267270787362443, 6.741833807488392010474612375131, 7.2910239781827480007377331501, 8.09253809277852589680758481236, 8.533566155795273367107074111740, 9.713617501296990007035103411403, 10.270256518447254738197778324768, 11.64332236070617208332652590836, 12.565765266378840226164179040540, 13.36434658349775983485549808508, 13.86042797258528128157588738567, 14.48624218219119770947948647249, 15.332609021058602660057131712078, 15.986349554731459734980134392394, 17.04227184016684900182343893749, 17.68378970380212503325690115627, 18.173548941429850354990782477014, 19.16898934728937025204345530241, 19.77146184882832899451618338062, 20.716060011544915815632428221572

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