↑

Properties

Label 1-760-760.597-r0-0-0
Degree $1$
Conductor $760$
Sign $-0.127 + 0.991i$
Analytic cond. $3.52942$
Root an. cond. $3.52942$
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  + (−0.866 + 0.5i)3-s + i·7-s + (0.5 − 0.866i)9-s − 11-s + (0.866 + 0.5i)13-s + (0.866 − 0.5i)17-s + (−0.5 − 0.866i)21-s + (0.866 + 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − 31-s + (0.866 − 0.5i)33-s − i·37-s − 39-s + (0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + i·7-s + (0.5 − 0.866i)9-s − 11-s + (0.866 + 0.5i)13-s + (0.866 − 0.5i)17-s + (−0.5 − 0.866i)21-s + (0.866 + 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − 31-s + (0.866 − 0.5i)33-s − i·37-s − 39-s + (0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.127 + 0.991i$
Analytic conductor: \(3.52942\)
Root analytic conductor: \(3.52942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (597, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (0:\ ),\ -0.127 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6275337586 + 0.7133807137i\)
\(L(\frac12)\) \(\approx\) \(0.6275337586 + 0.7133807137i\)
\(L(1)\) \(\approx\) \(0.7741679747 + 0.2818989114i\)
\(L(1)\) \(\approx\) \(0.7741679747 + 0.2818989114i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + iT \)
11 \( 1 - T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 - iT \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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

−22.48569395902260504827004788440, −21.26458886798218771271399333552, −20.75882835923239409809913225558, −19.65680893854232745199424122983, −18.87125737930351496012998462766, −18.006001028674953841324824935011, −17.499072075213929120395756334496, −16.372377381120511962798618484029, −16.134002916290645815829309983731, −14.7938370783304680781080820680, −13.833188741695778461681444174400, −12.90070658744977391272193366798, −12.61272662357446956589562888693, −11.18531244467815918094811901634, −10.71865085523505972507200344712, −10.065026487966611314934089428607, −8.587218358752409438806400335446, −7.64060440535154382668679506957, −7.04996940971531596309562662144, −5.93161254976708452824951616037, −5.24072510355336029865548248171, −4.15891436124570254238445141278, −3.03281557118886712257514179344, −1.58715762442926598054583659303, −0.59531212268847674630530410959, 1.1572853253853268206204291732, 2.59849995832265760459346504029, 3.63804188135377230059892546405, 4.85187435323649471225070585608, 5.524098900669566841934619032184, 6.25190949240040548769190620371, 7.37257747796021771388347696221, 8.505805736035451347444217089194, 9.395173961033108886962980934675, 10.18345142040733668354635304474, 11.1847402820935994663795420943, 11.73701741168822553427971889516, 12.64314539817910023651642947556, 13.46977103614081860126366099151, 14.721401767115154712750931567403, 15.496210610844355743966295360894, 16.088878042964664828772946057760, 16.82646842140323932014923485199, 17.92108686751366071374221093299, 18.45265888444419193523604184764, 19.12893554185350659060178731920, 20.586742813355696901180364592376, 21.19797616079492700312792080773, 21.66605042961259898392943800614, 22.724738646457475840324919737403

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