Properties

Label 1-1375-1375.1097-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.927 + 0.373i$
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  + (−0.481 − 0.876i)2-s + (0.998 − 0.0627i)3-s + (−0.535 + 0.844i)4-s + (−0.535 − 0.844i)6-s + (−0.951 − 0.309i)7-s + (0.998 + 0.0627i)8-s + (0.992 − 0.125i)9-s + (−0.481 + 0.876i)12-s + (−0.125 − 0.992i)13-s + (0.187 + 0.982i)14-s + (−0.425 − 0.904i)16-s + (−0.248 − 0.968i)17-s + (−0.587 − 0.809i)18-s + (−0.929 − 0.368i)19-s + (−0.968 − 0.248i)21-s + ⋯
L(s)  = 1  + (−0.481 − 0.876i)2-s + (0.998 − 0.0627i)3-s + (−0.535 + 0.844i)4-s + (−0.535 − 0.844i)6-s + (−0.951 − 0.309i)7-s + (0.998 + 0.0627i)8-s + (0.992 − 0.125i)9-s + (−0.481 + 0.876i)12-s + (−0.125 − 0.992i)13-s + (0.187 + 0.982i)14-s + (−0.425 − 0.904i)16-s + (−0.248 − 0.968i)17-s + (−0.587 − 0.809i)18-s + (−0.929 − 0.368i)19-s + (−0.968 − 0.248i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.927 + 0.373i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.927 + 0.373i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1182934791 - 0.6103864499i\)
\(L(\frac12)\) \(\approx\) \(-0.1182934791 - 0.6103864499i\)
\(L(1)\) \(\approx\) \(0.6950277092 - 0.4540334580i\)
\(L(1)\) \(\approx\) \(0.6950277092 - 0.4540334580i\)

Euler product

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

−21.48207982207795839193055173926, −20.14106287906623655480639124808, −19.55638480831156991348340990862, −19.01953726258056584028993648849, −18.4215888955202797169660347271, −17.412386158642112086522198993529, −16.52086185345398730143493719744, −15.921386553408219003834454108361, −15.29709408843248300912637543016, −14.41012885929974507849646374434, −13.988838152710329866762665919125, −12.95931185416099063424108402573, −12.40303564750326976778703349262, −10.87499779810990693372511074159, −10.0322051476578706706001241061, −9.47559304007169291166661496308, −8.619755084220070718235891639425, −8.20662492942343294621293781372, −7.0079846445266475024818581775, −6.549425880391785508420880411000, −5.60876725762400035186155979170, −4.29735852951976284889558832672, −3.78119148407600023829255204557, −2.37580241954795193215999274866, −1.603475725715170382127031786023, 0.23472668790259190461446761029, 1.51449757762679540333548329812, 2.63113682775418610977275866951, 3.1107042767953243688439707462, 3.991255365896893371544654878770, 4.86812591647990732505370380106, 6.3968474392255057690575068758, 7.31112024729235509344034170778, 8.036613635406357864135377453724, 8.837511287204033854326985440508, 9.60118464075571451427966192514, 10.15443341738593537945783082136, 10.91054133106756125962044305410, 12.11457363320095521185587234546, 12.753426383870380917175235171297, 13.41908770456915087040001424171, 13.98443863930841952931075617637, 15.107351777985941543350835093162, 15.92210211871892310599778149863, 16.6359358835767411255122293012, 17.71759697135791332135201796536, 18.29987045459103477846199730532, 19.15642498120472109072776986521, 19.76270492416221339893644533188, 20.19845757624664000352818752116

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