↑

Properties

Label 1-647-647.338-r0-0-0
Degree $1$
Conductor $647$
Sign $0.539 - 0.842i$
Analytic cond. $3.00465$
Root an. cond. $3.00465$
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  + (−0.850 + 0.526i)2-s + (0.932 − 0.361i)3-s + (0.445 − 0.895i)4-s + (0.739 − 0.673i)5-s + (−0.602 + 0.798i)6-s + (0.0922 − 0.995i)7-s + (0.0922 + 0.995i)8-s + (0.739 − 0.673i)9-s + (−0.273 + 0.961i)10-s + (0.739 + 0.673i)11-s + (0.0922 − 0.995i)12-s + (0.0922 − 0.995i)13-s + (0.445 + 0.895i)14-s + (0.445 − 0.895i)15-s + (−0.602 − 0.798i)16-s + (0.932 − 0.361i)17-s + ⋯
L(s)  = 1  + (−0.850 + 0.526i)2-s + (0.932 − 0.361i)3-s + (0.445 − 0.895i)4-s + (0.739 − 0.673i)5-s + (−0.602 + 0.798i)6-s + (0.0922 − 0.995i)7-s + (0.0922 + 0.995i)8-s + (0.739 − 0.673i)9-s + (−0.273 + 0.961i)10-s + (0.739 + 0.673i)11-s + (0.0922 − 0.995i)12-s + (0.0922 − 0.995i)13-s + (0.445 + 0.895i)14-s + (0.445 − 0.895i)15-s + (−0.602 − 0.798i)16-s + (0.932 − 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(647\)
Sign: $0.539 - 0.842i$
Analytic conductor: \(3.00465\)
Root analytic conductor: \(3.00465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{647} (338, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 647,\ (0:\ ),\ 0.539 - 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.432112333 - 0.7833343309i\)
\(L(\frac12)\) \(\approx\) \(1.432112333 - 0.7833343309i\)
\(L(1)\) \(\approx\) \(1.171908590 - 0.2588598505i\)
\(L(1)\) \(\approx\) \(1.171908590 - 0.2588598505i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad647 \( 1 \)
good2 \( 1 + (-0.850 + 0.526i)T \)
3 \( 1 + (0.932 - 0.361i)T \)
5 \( 1 + (0.739 - 0.673i)T \)
7 \( 1 + (0.0922 - 0.995i)T \)
11 \( 1 + (0.739 + 0.673i)T \)
13 \( 1 + (0.0922 - 0.995i)T \)
17 \( 1 + (0.932 - 0.361i)T \)
19 \( 1 + (-0.602 - 0.798i)T \)
23 \( 1 + (-0.602 + 0.798i)T \)
29 \( 1 + (-0.602 + 0.798i)T \)
31 \( 1 + (0.739 + 0.673i)T \)
37 \( 1 + (-0.602 + 0.798i)T \)
41 \( 1 + (0.0922 + 0.995i)T \)
43 \( 1 + (-0.850 - 0.526i)T \)
47 \( 1 + (0.932 - 0.361i)T \)
53 \( 1 + (-0.602 + 0.798i)T \)
59 \( 1 + (-0.273 + 0.961i)T \)
61 \( 1 + (0.932 + 0.361i)T \)
67 \( 1 + (0.932 + 0.361i)T \)
71 \( 1 + (-0.602 + 0.798i)T \)
73 \( 1 + (-0.850 + 0.526i)T \)
79 \( 1 + (-0.850 - 0.526i)T \)
83 \( 1 + (-0.273 - 0.961i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (-0.273 + 0.961i)T \)
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

−22.473498004960615681905825565198, −21.882075235953541825613323677140, −21.1202146950811040342951585722, −20.82449193697345611369564498557, −19.39340265135723310985263051204, −18.87391565229945245992977960898, −18.56078490647459517323964234096, −17.25769791328059523849664395276, −16.53732052324321345169966224085, −15.59199044084840802240752406568, −14.54351888354054591408738082457, −14.05356302043091608123812179679, −12.86602509583970176512836264232, −11.9173777251222391101741300227, −10.99499628459736166523695880627, −10.03221579714473711111599650133, −9.43822663563150676158106274338, −8.6558030547090702882423168439, −7.95068018028984672264355314970, −6.690165649477834076720473983593, −5.85792562384328074615762237862, −4.10490212189799316863796898068, −3.29290372921967177755443152166, −2.22806848289658336996419662738, −1.72560259430328908853073238607, 1.04656176489028227776844662277, 1.63035323491496585482334367262, 2.96358483476695429441023941609, 4.36188867521235083416819859834, 5.46312757483904680667534273992, 6.66057765198916372243868164553, 7.340929223863208495582449553112, 8.22653220160391075560751143500, 8.99109709412711735473408637728, 9.899212105564716630881657926091, 10.31254377380819513042687316246, 11.82397307422383045729537628966, 12.929020150739576192701564529366, 13.73986218780208042256354151539, 14.42017038553389773057633850538, 15.26415080372981327722390603612, 16.22753917894667207631264477916, 17.28244649547728165074617683203, 17.56181943782577059571482666994, 18.55360760117453058360453971432, 19.60855137273741344098970206541, 20.23177925984047124411374883733, 20.549759386597531511223679153173, 21.72498277444695026689652387338, 23.16793187283988359193208945697

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