Properties

Label 1-1375-1375.652-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.307 - 0.951i$
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  + (0.998 + 0.0627i)2-s + (0.684 − 0.728i)3-s + (0.992 + 0.125i)4-s + (0.728 − 0.684i)6-s + (−0.951 + 0.309i)7-s + (0.982 + 0.187i)8-s + (−0.0627 − 0.998i)9-s + (0.770 − 0.637i)12-s + (0.998 − 0.0627i)13-s + (−0.968 + 0.248i)14-s + (0.968 + 0.248i)16-s + (−0.982 − 0.187i)17-s i·18-s + (0.425 + 0.904i)19-s + (−0.425 + 0.904i)21-s + ⋯
L(s)  = 1  + (0.998 + 0.0627i)2-s + (0.684 − 0.728i)3-s + (0.992 + 0.125i)4-s + (0.728 − 0.684i)6-s + (−0.951 + 0.309i)7-s + (0.982 + 0.187i)8-s + (−0.0627 − 0.998i)9-s + (0.770 − 0.637i)12-s + (0.998 − 0.0627i)13-s + (−0.968 + 0.248i)14-s + (0.968 + 0.248i)16-s + (−0.982 − 0.187i)17-s i·18-s + (0.425 + 0.904i)19-s + (−0.425 + 0.904i)21-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.307 - 0.951i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (652, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.307 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.669390188 - 3.668855934i\)
\(L(\frac12)\) \(\approx\) \(2.669390188 - 3.668855934i\)
\(L(1)\) \(\approx\) \(2.110751448 - 0.7080379127i\)
\(L(1)\) \(\approx\) \(2.110751448 - 0.7080379127i\)

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 \( 1 + (0.684 - 0.728i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (0.998 - 0.0627i)T \)
17 \( 1 + (-0.982 - 0.187i)T \)
19 \( 1 + (0.425 + 0.904i)T \)
23 \( 1 + (-0.998 - 0.0627i)T \)
29 \( 1 + (-0.728 - 0.684i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (-0.368 - 0.929i)T \)
41 \( 1 + (-0.637 - 0.770i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (0.125 - 0.992i)T \)
53 \( 1 + (0.684 - 0.728i)T \)
59 \( 1 + (-0.0627 - 0.998i)T \)
61 \( 1 + (0.968 - 0.248i)T \)
67 \( 1 + (-0.125 - 0.992i)T \)
71 \( 1 + (-0.425 + 0.904i)T \)
73 \( 1 + (-0.248 - 0.968i)T \)
79 \( 1 + (0.992 + 0.125i)T \)
83 \( 1 + (0.125 + 0.992i)T \)
89 \( 1 + (0.637 - 0.770i)T \)
97 \( 1 + (-0.904 - 0.425i)T \)
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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

−20.788323400628529628215028009752, −20.30227369963824305526341156845, −19.66278502393345965977000554965, −19.03463877127223307809335523288, −17.80074320490812960500358639940, −16.63796683846588380029972590401, −15.99966779078735048173178266220, −15.59115600628192004704269471161, −14.81188981187290954723724136405, −13.69106017341896828152186989535, −13.57837265993391227413244380375, −12.711496903728744355505569306508, −11.603809747103561800062898021674, −10.8139167821459667604819703244, −10.183536847645026671516274335786, −9.26264473379060271216078639631, −8.42415081901222287397508282830, −7.33560425679200780065607908734, −6.53134528572989475111640167888, −5.70158896499850971559953321408, −4.61925185701524932058183035740, −3.98233853731146599331130091041, −3.19767224955066646286324988513, −2.52009865611212447330256885764, −1.290365579423435307071247304750, 0.49065507995768129653338884005, 1.89159095240504360310399033529, 2.46336658458619638182497341077, 3.663201141510492857389252436095, 3.88542378065791872967569916899, 5.50760180531503928034955230928, 6.18471488662769898746752589338, 6.789523710751063801978034008888, 7.715786940945519528408889956521, 8.512016748743667237091281891993, 9.46913985833391985233349662123, 10.41584668719234116053639931557, 11.53929406308722158706066962863, 12.1758494206476467454257545842, 12.93319124497109665915414804705, 13.55790184078284234010861241667, 14.02643668625924801688158512366, 15.06617694381356743925816190843, 15.68818493083551815601295803918, 16.291569039927405745642318566607, 17.402332097199288669493984579680, 18.39111822468825563085052446944, 19.0525220255151706579218442449, 19.7947294237742616678948318646, 20.519382304300831846905293629305

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