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Properties

Label 1-1375-1375.1027-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.262 - 0.965i$
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  + (−0.844 + 0.535i)2-s + (0.904 + 0.425i)3-s + (0.425 − 0.904i)4-s + (−0.992 + 0.125i)6-s + i·7-s + (0.125 + 0.992i)8-s + (0.637 + 0.770i)9-s + (0.770 − 0.637i)12-s + (−0.770 + 0.637i)13-s + (−0.535 − 0.844i)14-s + (−0.637 − 0.770i)16-s + (0.684 + 0.728i)17-s + (−0.951 − 0.309i)18-s + (−0.728 + 0.684i)19-s + (−0.425 + 0.904i)21-s + ⋯
L(s)  = 1  + (−0.844 + 0.535i)2-s + (0.904 + 0.425i)3-s + (0.425 − 0.904i)4-s + (−0.992 + 0.125i)6-s + i·7-s + (0.125 + 0.992i)8-s + (0.637 + 0.770i)9-s + (0.770 − 0.637i)12-s + (−0.770 + 0.637i)13-s + (−0.535 − 0.844i)14-s + (−0.637 − 0.770i)16-s + (0.684 + 0.728i)17-s + (−0.951 − 0.309i)18-s + (−0.728 + 0.684i)19-s + (−0.425 + 0.904i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.262 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.262 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.262 - 0.965i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.262 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4266714785 + 0.5580414747i\)
\(L(\frac12)\) \(\approx\) \(-0.4266714785 + 0.5580414747i\)
\(L(1)\) \(\approx\) \(0.6580643442 + 0.5155525731i\)
\(L(1)\) \(\approx\) \(0.6580643442 + 0.5155525731i\)

Euler product

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

−20.1127111558976230885064198009, −19.37938623243505576003291941881, −18.83024222894234736921329140354, −17.87861585694078056527047522792, −17.34361058455393216324842576148, −16.51063944084879441240332703785, −15.61494721957229574424582721237, −14.74088129902307372622274627406, −13.858208864602616707616174657, −13.119631180568568665230464412253, −12.47029556569042398860653208002, −11.58446540735990220953725983330, −10.62100319529664409636611009173, −9.85568558179432067843105661845, −9.33449168494427880476996814105, −8.30207434553059922508268301022, −7.52274079587340350031900805166, −7.27220269462850846296375294570, −6.09564578957610434390163389638, −4.48578825379124294721698275744, −3.71157542069290552531332923243, −2.80466301437626817124322021282, −2.05560395034440313008743332220, −0.94939560387290030731666201068, −0.16462530167530419207753366709, 1.71586147326515505827397559631, 2.12334954996204916477118087119, 3.30213059286042973005948405431, 4.44132327259567016023575647061, 5.4401091325565599341433049972, 6.19867382235296755143276520471, 7.35439013189138749381086167710, 7.94826547180399997637065161841, 8.91798159522504934317830577659, 9.155767308988214876750037207378, 10.21642417080345266563133348818, 10.69218579320239114984652264278, 12.02026369837842683263228440791, 12.62020409084866538149568326323, 14.2189174452321097792327953845, 14.28867387844312962377283810497, 15.27523707556937299863899683818, 15.75701420732407330368739485797, 16.62968204811380295543337908566, 17.266658724531201513062456757068, 18.43533042014729619764643970389, 18.889573817859966084703352290457, 19.48663801695413634498661748512, 20.24387670106134416397544887722, 21.153921715485129481023050517567

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