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Properties

Label 1-1375-1375.312-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.310 + 0.950i$
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  + (−0.368 − 0.929i)2-s + (0.982 + 0.187i)3-s + (−0.728 + 0.684i)4-s + (−0.187 − 0.982i)6-s + (0.951 + 0.309i)7-s + (0.904 + 0.425i)8-s + (0.929 + 0.368i)9-s + (−0.844 + 0.535i)12-s + (−0.368 + 0.929i)13-s + (−0.0627 − 0.998i)14-s + (0.0627 − 0.998i)16-s + (−0.904 − 0.425i)17-s − i·18-s + (−0.876 + 0.481i)19-s + (0.876 + 0.481i)21-s + ⋯
L(s)  = 1  + (−0.368 − 0.929i)2-s + (0.982 + 0.187i)3-s + (−0.728 + 0.684i)4-s + (−0.187 − 0.982i)6-s + (0.951 + 0.309i)7-s + (0.904 + 0.425i)8-s + (0.929 + 0.368i)9-s + (−0.844 + 0.535i)12-s + (−0.368 + 0.929i)13-s + (−0.0627 − 0.998i)14-s + (0.0627 − 0.998i)16-s + (−0.904 − 0.425i)17-s − i·18-s + (−0.876 + 0.481i)19-s + (0.876 + 0.481i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.310 + 0.950i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (312, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.310 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.562000014 + 1.133238811i\)
\(L(\frac12)\) \(\approx\) \(1.562000014 + 1.133238811i\)
\(L(1)\) \(\approx\) \(1.184770016 - 0.1162899619i\)
\(L(1)\) \(\approx\) \(1.184770016 - 0.1162899619i\)

Euler product

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

−20.16191111968158420771511797176, −19.81889657718173870758722255526, −18.89200891135285285316999664534, −18.09014599599697156899404701933, −17.573172993447947275448615809548, −16.79830761137854125346734401298, −15.71660459024805701108258901701, −15.06541840424556925988840687753, −14.62386480410815354175073141563, −13.82985139304305950111130245950, −13.110649113190315982581760434226, −12.34274481324830289760764202996, −10.71254588296638254115065604549, −10.46676061663937839024554814365, −9.23821016919920079196992116092, −8.444245439315432728454201371158, −8.20986168013201384893643488218, −7.05835156595588017568103315591, −6.697225719818430720967758705781, −5.26121064116939809827565928670, −4.61973263336653126740916026997, −3.687489034686578686970726508743, −2.38072556750218016527521840575, −1.45557770622886321239291630291, −0.3564136125094280554456807136, 1.233683514835717334631019205992, 2.15691316514372331672189353663, 2.559307891020823310791324897501, 3.98866970614476253694034089151, 4.31160449803283929520688128725, 5.36432526282896143413694514014, 6.949512307096939514359413178845, 7.77540749590789916105056495684, 8.50171609160515218165745274630, 9.147245860419313401852033029860, 9.79003489358862933066097788209, 10.771455672474012394110839371967, 11.480024436784597257497707527630, 12.21032187145180083192714777964, 13.26197949356479953127777726477, 13.767387852931928214483147060625, 14.62423604305567806470473900743, 15.26857700067325986181382609101, 16.35121858233665764168780721215, 17.240177438141421123570707252258, 17.96610010861184649292236163377, 18.826958799038269602579619995776, 19.288070880352653279217380386540, 20.06851269746962847789161890846, 20.84894221339308790237988600933

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