↑

Properties

Label 1-725-725.91-r0-0-0
Degree $1$
Conductor $725$
Sign $0.992 - 0.123i$
Analytic cond. $3.36688$
Root an. cond. $3.36688$
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  + (−0.134 + 0.990i)2-s + (0.0448 − 0.998i)3-s + (−0.963 − 0.266i)4-s + (0.983 + 0.178i)6-s + (0.623 − 0.781i)7-s + (0.393 − 0.919i)8-s + (−0.995 − 0.0896i)9-s + (0.995 − 0.0896i)11-s + (−0.309 + 0.951i)12-s + (0.858 + 0.512i)13-s + (0.691 + 0.722i)14-s + (0.858 + 0.512i)16-s + (0.809 + 0.587i)17-s + (0.222 − 0.974i)18-s + (0.0448 + 0.998i)19-s + ⋯
L(s)  = 1  + (−0.134 + 0.990i)2-s + (0.0448 − 0.998i)3-s + (−0.963 − 0.266i)4-s + (0.983 + 0.178i)6-s + (0.623 − 0.781i)7-s + (0.393 − 0.919i)8-s + (−0.995 − 0.0896i)9-s + (0.995 − 0.0896i)11-s + (−0.309 + 0.951i)12-s + (0.858 + 0.512i)13-s + (0.691 + 0.722i)14-s + (0.858 + 0.512i)16-s + (0.809 + 0.587i)17-s + (0.222 − 0.974i)18-s + (0.0448 + 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.992 - 0.123i$
Analytic conductor: \(3.36688\)
Root analytic conductor: \(3.36688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 725,\ (0:\ ),\ 0.992 - 0.123i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.420746669 - 0.08802690200i\)
\(L(\frac12)\) \(\approx\) \(1.420746669 - 0.08802690200i\)
\(L(1)\) \(\approx\) \(1.078575161 + 0.05832765338i\)
\(L(1)\) \(\approx\) \(1.078575161 + 0.05832765338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.134 + 0.990i)T \)
3 \( 1 + (0.0448 - 0.998i)T \)
7 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (0.995 - 0.0896i)T \)
13 \( 1 + (0.858 + 0.512i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (0.0448 + 0.998i)T \)
23 \( 1 + (-0.691 - 0.722i)T \)
31 \( 1 + (-0.473 - 0.880i)T \)
37 \( 1 + (0.995 + 0.0896i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.393 + 0.919i)T \)
53 \( 1 + (0.983 - 0.178i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.936 - 0.351i)T \)
67 \( 1 + (-0.393 + 0.919i)T \)
71 \( 1 + (-0.393 - 0.919i)T \)
73 \( 1 + (0.691 + 0.722i)T \)
79 \( 1 + (-0.753 - 0.657i)T \)
83 \( 1 + (-0.0448 - 0.998i)T \)
89 \( 1 + (-0.134 + 0.990i)T \)
97 \( 1 + (0.963 + 0.266i)T \)
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

−22.33507522356204966167105080960, −21.499068769764913413709025329838, −21.146198063803638534111491454365, −20.10791656564678066259579505009, −19.691462078510243935338369429247, −18.467207733151148650672714245258, −17.828788169802494607145047235363, −17.012749332055577749584758014054, −16.01060031071363513460929304864, −15.10225214568710732366412967676, −14.28979681302719873697470735785, −13.5753365654568628917121592425, −12.25186196258262306077968307280, −11.622247909319564900798603186503, −10.968005250178687385858450597802, −10.05794619894690412148228913165, −9.097749385546014910235851842379, −8.759103819862381912463980143710, −7.6340668937513181212761115858, −5.86203392273893587842361424957, −5.16172270957230239088234440490, −4.16712994986968464281603612385, −3.35630698177884214568490057067, −2.398663150652514955109403551140, −1.11086211012864900978384933737, 0.95828585532364293358569791953, 1.70885508790034722891040068702, 3.629301006426744778816896691414, 4.35803216246332433446066249101, 5.85284016685214263045628420893, 6.27681777430120812641093102113, 7.30995853413832669848014398615, 8.000411894076685013776973760375, 8.6579131579549504016800733647, 9.74935447400773816057799043914, 10.880622677188465968773174701910, 11.875218378843755804206308833348, 12.8225052027961468768334046734, 13.73630704667069318660324718273, 14.34071065519856571509185508166, 14.81060438070059664859113720382, 16.38207918715349590384172602705, 16.79129897892214268050418137855, 17.56725851128283283523097897800, 18.4323817082470255164677487755, 18.95927844037206373276592237934, 19.92885280716258364224042590875, 20.79702824075851685116542926502, 22.01037404982980751113962389597, 22.94476753422313837171596885709

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