↑

Properties

Label 1-712-712.123-r1-0-0
Degree $1$
Conductor $712$
Sign $0.484 + 0.874i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + i·3-s + 5-s − i·7-s − 9-s + 11-s + i·13-s + i·15-s − 17-s − i·19-s + 21-s − i·23-s + 25-s − i·27-s + i·29-s + i·31-s + ⋯
L(s)  = 1  + i·3-s + 5-s − i·7-s − 9-s + 11-s + i·13-s + i·15-s − 17-s − i·19-s + 21-s − i·23-s + 25-s − i·27-s + i·29-s + i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.484 + 0.874i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.484 + 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.220777803 + 1.308208361i\)
\(L(\frac12)\) \(\approx\) \(2.220777803 + 1.308208361i\)
\(L(1)\) \(\approx\) \(1.280414686 + 0.3672181004i\)
\(L(1)\) \(\approx\) \(1.280414686 + 0.3672181004i\)

Euler product

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

−22.44159751430192942043688995666, −21.532342301285199405092220662, −20.57874712632287269884972628778, −19.72712133874989335385630800777, −18.887115247073944961059457188838, −18.15372661467601582514247587193, −17.47772497724173528196808176132, −16.92211312574122441527166434459, −15.53481218840709297445212981019, −14.743890932404893088333670615432, −13.837027364212927438658875685010, −13.18621839617173755130059571028, −12.32388481576868234151408969355, −11.67158542269749179772141887036, −10.575210793190574389697810736219, −9.391115656989898669606625448823, −8.812071098528341007159298654223, −7.80446508049826205013484739986, −6.74401790804547737114068330544, −5.8205660788427989906313550077, −5.52511988713944468118103881568, −3.784826068845011137640921535268, −2.445087683238729037901699735843, −1.93160692888387582631493599020, −0.709394880149757356330650973754, 0.93806301253532666700497496699, 2.20256417378804199143382190927, 3.39007552786935304398253451291, 4.4564817534221805569951566697, 4.95169468605166632349848389148, 6.52147724356626600561878143456, 6.7260394580654585607180893121, 8.51001917622657915430073441565, 9.20880416211231870393082924251, 9.85894370498987052515780383439, 10.82812977847148656325476583131, 11.33153109007821022705020724990, 12.66845214954948266208050547291, 13.830022933976472993723497436499, 14.13974425156632629493839837711, 15.041784543214563945577320612981, 16.20721235948912358833951552632, 16.83462224529508824246283595127, 17.364238010102089733645835292219, 18.26943771891813741845676572520, 19.68767940454165097883940985122, 20.10202958454503299555671559219, 20.99917259648902795563447100589, 21.85479162127098323135414798192, 22.18196178970433064618376889563

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