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Properties

Label 1-1375-1375.1048-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  + (0.844 − 0.535i)2-s + (0.481 + 0.876i)3-s + (0.425 − 0.904i)4-s + (0.876 + 0.481i)6-s + (0.951 − 0.309i)7-s + (−0.125 − 0.992i)8-s + (−0.535 + 0.844i)9-s + (0.998 − 0.0627i)12-s + (0.844 + 0.535i)13-s + (0.637 − 0.770i)14-s + (−0.637 − 0.770i)16-s + (0.125 + 0.992i)17-s + i·18-s + (−0.728 + 0.684i)19-s + (0.728 + 0.684i)21-s + ⋯
L(s)  = 1  + (0.844 − 0.535i)2-s + (0.481 + 0.876i)3-s + (0.425 − 0.904i)4-s + (0.876 + 0.481i)6-s + (0.951 − 0.309i)7-s + (−0.125 − 0.992i)8-s + (−0.535 + 0.844i)9-s + (0.998 − 0.0627i)12-s + (0.844 + 0.535i)13-s + (0.637 − 0.770i)14-s + (−0.637 − 0.770i)16-s + (0.125 + 0.992i)17-s + i·18-s + (−0.728 + 0.684i)19-s + (0.728 + 0.684i)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: $\chi_{1375} (1048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.307 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.472284682 + 2.526368662i\)
\(L(\frac12)\) \(\approx\) \(3.472284682 + 2.526368662i\)
\(L(1)\) \(\approx\) \(2.081704955 + 0.1823282086i\)
\(L(1)\) \(\approx\) \(2.081704955 + 0.1823282086i\)

Euler product

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

−20.570051171903457526050863592230, −20.06991659373228657790095027556, −18.78487694753011694712662073453, −18.21698670474709761634381880733, −17.47775907717584127461660908287, −16.77103554520925382664880882990, −15.517755478281241221657383692673, −15.18155452633888441073248168039, −14.24700618252073550018713964304, −13.660343269347881439597427948970, −13.08985089238889642628455936042, −12.09177050761987513517831007018, −11.63187109292503498476116339870, −10.74078933559267535637646041030, −9.23561427257572068100489265440, −8.31723223670971914217272272941, −7.98572916804805394375128422253, −7.024079153945237279950667655935, −6.24936015857014573309848435941, −5.465790229737509293290723738293, −4.52150444255428149861534492207, −3.54043493965181301332822941464, −2.55195503690990230521652970770, −1.875873600286917110046185568828, −0.50005097500593279947886524437, 1.37408738638302744276345814062, 2.004811733105786526160179333549, 3.19450021511024075143614216151, 4.10977567250587180032043295176, 4.37295599356021074412791255996, 5.54749150411298891119481589592, 6.16996019071896225681296387164, 7.52838071904341072597832308974, 8.39636778504024821430834510260, 9.221297721158323885393844084742, 10.2955878257807679776733928752, 10.71508586277080640133333198275, 11.451635416822129960375421554700, 12.28954918896915688992865399190, 13.42504080350649705236077165176, 13.90298687765576057885910243868, 14.78372204259917497692475186335, 15.06585823240402491006316023675, 16.139800505571855190810931226966, 16.72935618837712927674931105591, 17.856047213003044291711694892493, 18.81919483685495386089062542475, 19.59105973947891332447574045678, 20.279333853542561556002446225780, 20.961467132692306575071499479101

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