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Properties

Label 1-1375-1375.1106-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.892 - 0.451i$
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  + (−0.876 − 0.481i)2-s + (0.968 − 0.248i)3-s + (0.535 + 0.844i)4-s + (−0.968 − 0.248i)6-s + (0.809 + 0.587i)7-s + (−0.0627 − 0.998i)8-s + (0.876 − 0.481i)9-s + (0.728 + 0.684i)12-s + (−0.876 + 0.481i)13-s + (−0.425 − 0.904i)14-s + (−0.425 + 0.904i)16-s + (−0.0627 − 0.998i)17-s − 18-s + (0.929 − 0.368i)19-s + (0.929 + 0.368i)21-s + ⋯
L(s)  = 1  + (−0.876 − 0.481i)2-s + (0.968 − 0.248i)3-s + (0.535 + 0.844i)4-s + (−0.968 − 0.248i)6-s + (0.809 + 0.587i)7-s + (−0.0627 − 0.998i)8-s + (0.876 − 0.481i)9-s + (0.728 + 0.684i)12-s + (−0.876 + 0.481i)13-s + (−0.425 − 0.904i)14-s + (−0.425 + 0.904i)16-s + (−0.0627 − 0.998i)17-s − 18-s + (0.929 − 0.368i)19-s + (0.929 + 0.368i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.892 - 0.451i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.892 - 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.500203036 - 0.5961007321i\)
\(L(\frac12)\) \(\approx\) \(2.500203036 - 0.5961007321i\)
\(L(1)\) \(\approx\) \(1.173874698 - 0.2357098671i\)
\(L(1)\) \(\approx\) \(1.173874698 - 0.2357098671i\)

Euler product

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

−20.49615165264747059959626779114, −19.96593934948953456740540240101, −19.136397670273016853058788134988, −18.57659763192900058436752346861, −17.53047365766696053521721315261, −17.05105894874210047294226342473, −16.2057682614036959319110930861, −15.12869443524532546306940182613, −14.93482078749370659692052799005, −14.07803079062644133265435883391, −13.34667943639218323037674479782, −12.16809702054773698209040387804, −11.12801443785390665904149442943, −10.27536805533846695965117605785, −9.88603400889281388182499025697, −8.786748542157215515684663804568, −8.24591495513099351522469610632, −7.47207832156491135290980425375, −6.96068657350105624451580347189, −5.57146202017700282123951155644, −4.799614546282452360348982079519, −3.74892698133199054705701722898, −2.583712803563042275199188694183, −1.722210814603631905948245973327, −0.77560042677626181072416961480, 0.79307228797137935719924073713, 1.7429157140915861089004103504, 2.52960332105837525777410278855, 3.19747841831468035025070665508, 4.375673253220266553079765142305, 5.35762181315221777065581536060, 6.972793112736395797662326396371, 7.32320713160981885477919446667, 8.18339180528233823099931549802, 9.08263306479925384560502069105, 9.368082468717171456616630523966, 10.34830164386277888478114558920, 11.476083870874375681368284755228, 11.91994963165983912128773325437, 12.80043110391302874809339455490, 13.73640705335516361509096173590, 14.46850864974300096108202666566, 15.39459610873806236361650554847, 15.93870432472968152447696848113, 17.08806986918858339337371129052, 17.78640354539473937694578404042, 18.46073650342340863390241107077, 19.11961311689444182874454383117, 19.69623950765438890607240213520, 20.702973856527072012368336386355

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