↑

Properties

Label 1-700-700.27-r1-0-0
Degree $1$
Conductor $700$
Sign $-0.904 - 0.425i$
Analytic cond. $75.2254$
Root an. cond. $75.2254$
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  + (−0.587 + 0.809i)3-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.951 + 0.309i)13-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (0.951 + 0.309i)23-s + (0.951 + 0.309i)27-s + (0.809 + 0.587i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.951 + 0.309i)37-s + (0.309 − 0.951i)39-s + (−0.309 − 0.951i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)3-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.951 + 0.309i)13-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (0.951 + 0.309i)23-s + (0.951 + 0.309i)27-s + (0.809 + 0.587i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.951 + 0.309i)37-s + (0.309 − 0.951i)39-s + (−0.309 − 0.951i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.904 - 0.425i$
Analytic conductor: \(75.2254\)
Root analytic conductor: \(75.2254\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 700,\ (1:\ ),\ -0.904 - 0.425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1296355703 + 0.5799562031i\)
\(L(\frac12)\) \(\approx\) \(-0.1296355703 + 0.5799562031i\)
\(L(1)\) \(\approx\) \(0.7002216116 + 0.3168982325i\)
\(L(1)\) \(\approx\) \(0.7002216116 + 0.3168982325i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (0.951 + 0.309i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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

−22.19460122274725361812967518944, −21.234025956925043888141774340639, −20.2958011142429839764645978974, −19.30033455108549123175032909458, −18.71279355839713360568500257114, −18.000719556541686829567933721345, −17.012153275254054033885559754837, −16.48924401069085673129055249443, −15.51579416908829501386562289102, −14.26914729405129252794825902503, −13.71298653722770301250261079585, −12.70403576189258362380113383851, −12.031328698759576319897859392623, −11.21899336503485192733538313494, −10.35821491019670282754607291195, −9.31895962300767286135662691619, −8.090403981873787830509776133988, −7.49124415752234081989626687275, −6.537877418473179193925737131294, −5.509203302370554454324065316813, −4.95632973960357207195266189592, −3.32164434062464370848738004887, −2.4089958893536453045128006740, −1.06530125245102680640275779962, −0.17371736320188675981616140027, 1.3173990022420622140224515184, 2.76217988300513092315318125928, 3.80144059535090977129620811155, 4.92760886562540032143653852792, 5.327207062336572212689609852056, 6.69434649000940516519067771444, 7.40178349480820760217744804141, 8.75082791080102608304412619344, 9.60916543727997242251447755491, 10.27910361910100608354502608182, 11.107107833995011410274339905, 12.15207305678049467225193272462, 12.62359736339158811827551464581, 14.00793783539654585438333340518, 14.8732921715615839154410692563, 15.488319749634271475125056015023, 16.366045730532638923410205016721, 17.26202227946889704634456367788, 17.69504343134289625438157396972, 18.80289745637095717050970312386, 19.8427246881585614324176913823, 20.5177741342354154954665648697, 21.50518968563508515210436572670, 21.9430915379869745796859067864, 22.92852766179401966632886418040

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