↑

Properties

Label 1-1375-1375.1147-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.703 - 0.710i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.904 − 0.425i)2-s + (0.248 − 0.968i)3-s + (0.637 + 0.770i)4-s + (−0.637 + 0.770i)6-s + (0.951 + 0.309i)7-s + (−0.248 − 0.968i)8-s + (−0.876 − 0.481i)9-s + (0.904 − 0.425i)12-s + (0.481 − 0.876i)13-s + (−0.728 − 0.684i)14-s + (−0.187 + 0.982i)16-s + (−0.844 + 0.535i)17-s + (0.587 + 0.809i)18-s + (−0.0627 + 0.998i)19-s + (0.535 − 0.844i)21-s + ⋯
L(s)  = 1  + (−0.904 − 0.425i)2-s + (0.248 − 0.968i)3-s + (0.637 + 0.770i)4-s + (−0.637 + 0.770i)6-s + (0.951 + 0.309i)7-s + (−0.248 − 0.968i)8-s + (−0.876 − 0.481i)9-s + (0.904 − 0.425i)12-s + (0.481 − 0.876i)13-s + (−0.728 − 0.684i)14-s + (−0.187 + 0.982i)16-s + (−0.844 + 0.535i)17-s + (0.587 + 0.809i)18-s + (−0.0627 + 0.998i)19-s + (0.535 − 0.844i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.703 - 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.497814879 - 0.6247402931i\)
\(L(\frac12)\) \(\approx\) \(1.497814879 - 0.6247402931i\)
\(L(1)\) \(\approx\) \(0.8068818487 - 0.3410952989i\)
\(L(1)\) \(\approx\) \(0.8068818487 - 0.3410952989i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−20.63098526572446533971387222120, −20.04797714404699434006802430888, −19.24551158882722435505759081532, −18.42091641268105711249044203699, −17.38801777026484795503005385717, −17.17970717690030131857478380235, −16.11947182284656220676243280424, −15.55185097572847761784278415496, −14.92851156687055642906475562597, −14.0440522404391169556315832406, −13.528466475938305075979491576299, −11.73535263153740089795658710429, −11.194208063849492865880093166824, −10.70999954343857548558101599954, −9.66720192927569255832884111300, −8.9838879569359270877156817617, −8.50115151498596707652106158571, −7.46818512368725360090983055517, −6.74551612796239257054074298243, −5.55799952248714922977410447253, −4.83251481347979500506855464164, −4.01305948957322168582089858044, −2.67242759528345523894590415462, −1.801258750164504800806024220121, −0.56014064427267742381576983512, 0.743049327456896523607221112860, 1.53260732145041358350808572302, 2.33258014336188782057491144562, 3.182418057714087173543365489116, 4.33026381059173094338203654411, 5.76789085447309878709404302989, 6.43499954954615407676125334930, 7.58093566191618754835045736850, 8.018382002696209971728448144397, 8.69389464213832296807091646347, 9.450916874101109478522137094106, 10.822781391565841293027893329630, 11.05481161585401542420459785973, 12.100216092436303738676888339534, 12.74202256960277086253468872778, 13.37317570694806706216035705712, 14.62093186863622796516155597588, 15.08570619014082134564427186073, 16.2345488107890953573018204288, 17.15783539177513524119532706572, 17.708274639829634888759968871607, 18.42296819111572205610206791408, 18.793339023369773793295470043168, 19.79096220956724644960349169840, 20.444979888560104783407676894974

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