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Properties

Label 1-1375-1375.1121-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.693 - 0.720i$
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  + (0.992 + 0.125i)2-s + (−0.637 − 0.770i)3-s + (0.968 + 0.248i)4-s + (−0.535 − 0.844i)6-s + (−0.309 − 0.951i)7-s + (0.929 + 0.368i)8-s + (−0.187 + 0.982i)9-s + (−0.425 − 0.904i)12-s + (0.187 − 0.982i)13-s + (−0.187 − 0.982i)14-s + (0.876 + 0.481i)16-s + (−0.968 + 0.248i)17-s + (−0.309 + 0.951i)18-s + (0.637 − 0.770i)19-s + (−0.535 + 0.844i)21-s + ⋯
L(s)  = 1  + (0.992 + 0.125i)2-s + (−0.637 − 0.770i)3-s + (0.968 + 0.248i)4-s + (−0.535 − 0.844i)6-s + (−0.309 − 0.951i)7-s + (0.929 + 0.368i)8-s + (−0.187 + 0.982i)9-s + (−0.425 − 0.904i)12-s + (0.187 − 0.982i)13-s + (−0.187 − 0.982i)14-s + (0.876 + 0.481i)16-s + (−0.968 + 0.248i)17-s + (−0.309 + 0.951i)18-s + (0.637 − 0.770i)19-s + (−0.535 + 0.844i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.693 - 0.720i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.693 - 0.720i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104850317 - 2.597820279i\)
\(L(\frac12)\) \(\approx\) \(1.104850317 - 2.597820279i\)
\(L(1)\) \(\approx\) \(1.457192937 - 0.5901133579i\)
\(L(1)\) \(\approx\) \(1.457192937 - 0.5901133579i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.992 + 0.125i)T \)
3 \( 1 + (-0.637 - 0.770i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.187 - 0.982i)T \)
17 \( 1 + (-0.968 + 0.248i)T \)
19 \( 1 + (0.637 - 0.770i)T \)
23 \( 1 + (0.728 + 0.684i)T \)
29 \( 1 + (-0.0627 - 0.998i)T \)
31 \( 1 + (0.968 - 0.248i)T \)
37 \( 1 + (0.876 + 0.481i)T \)
41 \( 1 + (-0.728 + 0.684i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (-0.929 + 0.368i)T \)
53 \( 1 + (0.535 - 0.844i)T \)
59 \( 1 + (-0.425 - 0.904i)T \)
61 \( 1 + (-0.728 - 0.684i)T \)
67 \( 1 + (0.0627 - 0.998i)T \)
71 \( 1 + (-0.929 + 0.368i)T \)
73 \( 1 + (0.425 - 0.904i)T \)
79 \( 1 + (0.637 + 0.770i)T \)
83 \( 1 + (0.637 - 0.770i)T \)
89 \( 1 + (-0.425 + 0.904i)T \)
97 \( 1 + (0.0627 + 0.998i)T \)
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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

−21.21929640227674108212296627410, −20.49093726765063943053835393825, −19.619980026571154523879385078, −18.68600195005596037221552423630, −17.94207714832389955549949405151, −16.7183494270249382096720634206, −16.25575207441324933565644396898, −15.59111992382998183905216740175, −14.8883971002303895899902813664, −14.19582119869578216523684158095, −13.2000539740027816803749294009, −12.30597770002594527825329556698, −11.79575001745655729031385984195, −11.09182790020911975937824933611, −10.30161838523281549892551137368, −9.35930281983363590945069818717, −8.6609561126762303006703227126, −7.14556506292313297364426094857, −6.3896059808995107627588336027, −5.7563051859311348568562535845, −4.88435060369996024128361896474, −4.256500585282012200962119676521, −3.26046355605058246002432482501, −2.43986145071902577271525504792, −1.1972377879105677770724469468, 0.4191553076648863186995577465, 1.341161408177462482729725075519, 2.53115490114322833111717085728, 3.3822411817262039241736587696, 4.51619779618399293236093098113, 5.1633926767601446413576657397, 6.21866073090550504146480104215, 6.68203143690563269035249872728, 7.58293963486219206298836546371, 8.12297544288245460729975554209, 9.697139521562578009865675026283, 10.72424661955624171481976676624, 11.20701653207162028961945358508, 11.98013144635708772028468152619, 13.02796535167948134444576285161, 13.321877179240939964145801175357, 13.86860113432112060212126866228, 15.10812931722373380316523002001, 15.7104576766734158050033412460, 16.599597346992937746917876045136, 17.34300221996665947061880631726, 17.789946509458395598795986000043, 19.05840633878622735899075013788, 19.786687689513293120280333146633, 20.28013293951040330181398896245

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