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Properties

Label 1-380-380.227-r1-0-0
Degree $1$
Conductor $380$
Sign $0.850 + 0.525i$
Analytic cond. $40.8366$
Root an. cond. $40.8366$
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  − i·3-s − i·7-s − 9-s − 11-s + i·13-s + i·17-s − 21-s + i·23-s + i·27-s + 29-s + 31-s + i·33-s − i·37-s + 39-s − 41-s + ⋯
L(s)  = 1  − i·3-s − i·7-s − 9-s − 11-s + i·13-s + i·17-s − 21-s + i·23-s + i·27-s + 29-s + 31-s + i·33-s − i·37-s + 39-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(40.8366\)
Root analytic conductor: \(40.8366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 380,\ (1:\ ),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076746606 + 0.3058811465i\)
\(L(\frac12)\) \(\approx\) \(1.076746606 + 0.3058811465i\)
\(L(1)\) \(\approx\) \(0.8872725076 - 0.2094566263i\)
\(L(1)\) \(\approx\) \(0.8872725076 - 0.2094566263i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

−24.37240463656695504097299819068, −23.08538148998178102316777965940, −22.47390616470362002979317647077, −21.62927781380062761239658485867, −20.78541096188507422055211926428, −20.20670916148701101144377857153, −18.90178388674697182804330827415, −18.10364311499436029008372343090, −17.16169983411012461202488474772, −15.8880578531410780927445147250, −15.62730782377802465031168511064, −14.683927967484963425192247855010, −13.615849761025835443544812881281, −12.45141610398539712745484804013, −11.570823747884138182475543713154, −10.47714402155996228231880545157, −9.83536792009014357753668110662, −8.67245382594252661365655406390, −8.02774222645886969018697409910, −6.402938466778331048840646255223, −5.29552324582213053609393702746, −4.756566635747933732155791353904, −3.13454769864309824474584108248, −2.53285296626793703180326361915, −0.35166839167439624994601327648, 1.03403719670332481660453019984, 2.12738247460264372583628007959, 3.43490222525270259821992509657, 4.70435748750243661352031037816, 5.994318778429037793335631643629, 6.92474696228286967913612152019, 7.74303367325734781462442089019, 8.59271611841181568235891590522, 9.97062951619419209722344805171, 10.92135182918894488977690084412, 11.86204737566106742688883983955, 12.90196065515974056743782956978, 13.61510873257668584605715418645, 14.302741733374395646467385780095, 15.55657888041836122892223377679, 16.69021328664430217859330246211, 17.41145798182367960950997378147, 18.24807271130062615206343280722, 19.25602876097835217788755393193, 19.7805504620641128595481171565, 20.88437654138870778751128090573, 21.714154010136134419374241137587, 23.2044702759179545045264876435, 23.44709429752069951647165780840, 24.21968702520285144942333376564

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