↑

Properties

Label 1-4235-4235.537-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.656 + 0.754i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.730 − 0.683i)2-s + (−0.743 + 0.669i)3-s + (0.0665 − 0.997i)4-s + (−0.0855 + 0.996i)6-s + (−0.633 − 0.774i)8-s + (0.104 − 0.994i)9-s + (0.618 + 0.786i)12-s + (−0.884 − 0.466i)13-s + (−0.991 − 0.132i)16-s + (−0.318 + 0.948i)17-s + (−0.603 − 0.797i)18-s + (−0.595 − 0.803i)19-s + (0.690 − 0.723i)23-s + (0.988 + 0.151i)24-s + (−0.964 + 0.263i)26-s + (0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.730 − 0.683i)2-s + (−0.743 + 0.669i)3-s + (0.0665 − 0.997i)4-s + (−0.0855 + 0.996i)6-s + (−0.633 − 0.774i)8-s + (0.104 − 0.994i)9-s + (0.618 + 0.786i)12-s + (−0.884 − 0.466i)13-s + (−0.991 − 0.132i)16-s + (−0.318 + 0.948i)17-s + (−0.603 − 0.797i)18-s + (−0.595 − 0.803i)19-s + (0.690 − 0.723i)23-s + (0.988 + 0.151i)24-s + (−0.964 + 0.263i)26-s + (0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.656 + 0.754i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1417654541 - 0.3111996361i\)
\(L(\frac12)\) \(\approx\) \(-0.1417654541 - 0.3111996361i\)
\(L(1)\) \(\approx\) \(0.8682481765 - 0.3837927733i\)
\(L(1)\) \(\approx\) \(0.8682481765 - 0.3837927733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.730 - 0.683i)T \)
3 \( 1 + (-0.743 + 0.669i)T \)
13 \( 1 + (-0.884 - 0.466i)T \)
17 \( 1 + (-0.318 + 0.948i)T \)
19 \( 1 + (-0.595 - 0.803i)T \)
23 \( 1 + (0.690 - 0.723i)T \)
29 \( 1 + (0.736 - 0.676i)T \)
31 \( 1 + (-0.272 - 0.962i)T \)
37 \( 1 + (-0.768 + 0.640i)T \)
41 \( 1 + (-0.516 - 0.856i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 + (0.603 - 0.797i)T \)
53 \( 1 + (0.132 + 0.991i)T \)
59 \( 1 + (0.483 + 0.875i)T \)
61 \( 1 + (0.290 + 0.956i)T \)
67 \( 1 + (0.945 - 0.327i)T \)
71 \( 1 + (-0.870 - 0.491i)T \)
73 \( 1 + (0.0760 + 0.997i)T \)
79 \( 1 + (-0.449 - 0.893i)T \)
83 \( 1 + (0.336 + 0.941i)T \)
89 \( 1 + (-0.995 + 0.0950i)T \)
97 \( 1 + (-0.931 + 0.362i)T \)
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

−18.68867378537408667058856800123, −17.774039947567727382159470892041, −17.48667401218149328289671811431, −16.68348660653455941413183450523, −16.15981629329365872663409808682, −15.5723723112421187602307903038, −14.41085806004913006517381620348, −14.252467514705187937664937558038, −13.30678346670014910468185509151, −12.67563362295485559811531528970, −12.19741661518307490922746310227, −11.49324599296704739330289763942, −10.89986948792694510574918262148, −9.86994935075006391436488299423, −8.95172768523275362418789990389, −8.14118999752100388551161976524, −7.38664848611336329568364593269, −6.8579584703309155471727055733, −6.34006474890571724843098104410, −5.28711432573554368822913436754, −5.02706841020181885716680131254, −4.16248619035744490369682706574, −3.108917327091691751287769433795, −2.32066076723864505923484524292, −1.39380273698314470702876027027, 0.087115838517446204319055395068, 1.013977769991356331568569368963, 2.24514243207094125319766222292, 2.85816502005204175000162546612, 3.97084079188702034116697824735, 4.3273598586565182901022117938, 5.19019782720378496771943479908, 5.69479208710323434093196614212, 6.564850000695792307185593415536, 7.11920819344216817683757609028, 8.513615460833921375796345297380, 9.16038928864679986546284177904, 10.06248177101090498477899450882, 10.498998601707402637656907358987, 11.01924453747517898029086142135, 11.909283741092556376283721678403, 12.32830252989697215937962518174, 13.043988614847111542766523147475, 13.75020099630649457712209752401, 14.75374833737894019788435679735, 15.215748104156305280313125266742, 15.56669886804069622633542973234, 16.67767858579047570098031151165, 17.25391704036711018010639263305, 17.8143816026720276930256305065

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