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Properties

Label 1-1375-1375.112-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.964 + 0.263i$
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 + (0.982 + 0.187i)3-s + (0.992 − 0.125i)4-s + (0.992 + 0.125i)6-s + (−0.587 + 0.809i)7-s + (0.982 − 0.187i)8-s + (0.929 + 0.368i)9-s + (0.998 + 0.0627i)12-s + (0.368 − 0.929i)13-s + (−0.535 + 0.844i)14-s + (0.968 − 0.248i)16-s + (0.684 − 0.728i)17-s + (0.951 + 0.309i)18-s + (−0.425 + 0.904i)19-s + (−0.728 + 0.684i)21-s + ⋯
L(s)  = 1  + (0.998 − 0.0627i)2-s + (0.982 + 0.187i)3-s + (0.992 − 0.125i)4-s + (0.992 + 0.125i)6-s + (−0.587 + 0.809i)7-s + (0.982 − 0.187i)8-s + (0.929 + 0.368i)9-s + (0.998 + 0.0627i)12-s + (0.368 − 0.929i)13-s + (−0.535 + 0.844i)14-s + (0.968 − 0.248i)16-s + (0.684 − 0.728i)17-s + (0.951 + 0.309i)18-s + (−0.425 + 0.904i)19-s + (−0.728 + 0.684i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.964 + 0.263i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.964 + 0.263i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.423839646 + 0.5942082185i\)
\(L(\frac12)\) \(\approx\) \(4.423839646 + 0.5942082185i\)
\(L(1)\) \(\approx\) \(2.673657216 + 0.1947219605i\)
\(L(1)\) \(\approx\) \(2.673657216 + 0.1947219605i\)

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 \( 1 + (0.982 + 0.187i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + (0.368 - 0.929i)T \)
17 \( 1 + (0.684 - 0.728i)T \)
19 \( 1 + (-0.425 + 0.904i)T \)
23 \( 1 + (-0.844 - 0.535i)T \)
29 \( 1 + (0.728 - 0.684i)T \)
31 \( 1 + (-0.425 + 0.904i)T \)
37 \( 1 + (0.998 + 0.0627i)T \)
41 \( 1 + (-0.968 + 0.248i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (0.684 + 0.728i)T \)
53 \( 1 + (0.125 + 0.992i)T \)
59 \( 1 + (-0.968 + 0.248i)T \)
61 \( 1 + (-0.0627 - 0.998i)T \)
67 \( 1 + (-0.904 - 0.425i)T \)
71 \( 1 + (-0.992 + 0.125i)T \)
73 \( 1 + (-0.368 - 0.929i)T \)
79 \( 1 + (0.876 + 0.481i)T \)
83 \( 1 + (0.481 + 0.876i)T \)
89 \( 1 + (-0.0627 - 0.998i)T \)
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

−20.74896912083657306402876248886, −20.183504176151235351088852475056, −19.43943351709895644265238570816, −18.99348025110209775045863954527, −17.74553989817061148386839578543, −16.674424376185645615110902760286, −16.1518314721558700903244760872, −15.25514150638106525484399379865, −14.59291895000854533778154458774, −13.77508628417612769697660205566, −13.40825577402088317562074293019, −12.6228815759085569785773867972, −11.81576456811993206606976286757, −10.75608981810156199887543495067, −10.03379633999862089515007121029, −9.08291184456105107441156250540, −8.07642591166033380622956230157, −7.26295013438899593165554602836, −6.66016595888235527507621984832, −5.80303579570727950544995775635, −4.408450122989358231988154226115, −3.939434179612079039012958713419, −3.15524671067069649220509987086, −2.1844217721613047209103477486, −1.26088975358003803030830096992, 1.370094905202824353935279774217, 2.522736285492670936372442017258, 3.01525850868362307479645594212, 3.84536737279088997309693064033, 4.77110959646040836145022813516, 5.77731547883592039117633752542, 6.41058277195363517521991404829, 7.6202918774025246065626633477, 8.16444596009589345135142988592, 9.24471537614232519672240860822, 10.1059921682596737146785704724, 10.72001472210032040449273455132, 12.10012112517230415520139603960, 12.43949316035507372069484849676, 13.30718083117707873534641355493, 14.03686502651519504074636000648, 14.69839496909407153741855103191, 15.4472772367378485861983986847, 15.98383126620535158628169124152, 16.6547163688731194386170553782, 18.140823280069744220913203078575, 18.81447800646860494860894603846, 19.5834832671053008259868565577, 20.29088489134613239700193925327, 20.89126406607246540467551281863

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