Properties

Label 1-4235-4235.244-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.997 + 0.0752i$
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  + (−0.998 − 0.0570i)2-s + (0.309 − 0.951i)3-s + (0.993 + 0.113i)4-s + (−0.362 + 0.931i)6-s + (−0.985 − 0.170i)8-s + (−0.809 − 0.587i)9-s + (0.415 − 0.909i)12-s + (0.870 − 0.491i)13-s + (0.974 + 0.226i)16-s + (−0.941 + 0.336i)17-s + (0.774 + 0.633i)18-s + (−0.0285 + 0.999i)19-s + (0.654 + 0.755i)23-s + (−0.466 + 0.884i)24-s + (−0.897 + 0.441i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.998 − 0.0570i)2-s + (0.309 − 0.951i)3-s + (0.993 + 0.113i)4-s + (−0.362 + 0.931i)6-s + (−0.985 − 0.170i)8-s + (−0.809 − 0.587i)9-s + (0.415 − 0.909i)12-s + (0.870 − 0.491i)13-s + (0.974 + 0.226i)16-s + (−0.941 + 0.336i)17-s + (0.774 + 0.633i)18-s + (−0.0285 + 0.999i)19-s + (0.654 + 0.755i)23-s + (−0.466 + 0.884i)24-s + (−0.897 + 0.441i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.997 + 0.0752i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.997 + 0.0752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9989431018 + 0.03762517264i\)
\(L(\frac12)\) \(\approx\) \(0.9989431018 + 0.03762517264i\)
\(L(1)\) \(\approx\) \(0.7315807606 - 0.1778819978i\)
\(L(1)\) \(\approx\) \(0.7315807606 - 0.1778819978i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.998 + 0.0570i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.870 + 0.491i)T \)
17 \( 1 + (0.941 - 0.336i)T \)
19 \( 1 + (0.0285 - 0.999i)T \)
23 \( 1 + (-0.654 - 0.755i)T \)
29 \( 1 + (-0.564 - 0.825i)T \)
31 \( 1 + (0.198 - 0.980i)T \)
37 \( 1 + (-0.736 + 0.676i)T \)
41 \( 1 + (0.254 + 0.967i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (-0.774 + 0.633i)T \)
53 \( 1 + (0.974 - 0.226i)T \)
59 \( 1 + (-0.254 + 0.967i)T \)
61 \( 1 + (0.998 - 0.0570i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.610 + 0.791i)T \)
73 \( 1 + (0.516 - 0.856i)T \)
79 \( 1 + (0.696 - 0.717i)T \)
83 \( 1 + (0.0855 - 0.996i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.466 - 0.884i)T \)
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

−18.35234501065320933262414282185, −17.633746835572055345687398923846, −16.94243893182686028046358762041, −16.361828292633234548890383334586, −15.74505520727445571182656752983, −15.21195621861656752094841084228, −14.618565433585924859043870616120, −13.64776615402403730674653011788, −13.0565148046226800182370565776, −11.77106392565969619841121727347, −11.25333851724263107971376086011, −10.81999028210265450561200525849, −9.97523808352789332121174748382, −9.2960016499853808093056494936, −8.86024084243610293494860973116, −8.20232440840004702235526558651, −7.41884727005990098915407656033, −6.41578919925877348356470317942, −6.01510093040961006205826182751, −4.724364038870535486515640373906, −4.31092017856275396319593729993, −3.03971830428657693471591858320, −2.65467346182016987802880897358, −1.627441130127545043675832841, −0.44843847210910919205036800628, 0.860786630027220151649279629754, 1.56082139535162667040623236596, 2.25882535333264572768840336887, 3.210627842689612926326312256108, 3.769884839051375020905394651417, 5.31664612203245823083136416567, 6.03307079090479208159849901667, 6.70494526556882883153324139808, 7.348194144342144871055145432752, 8.027840315495780257187190226, 8.77823753055143796038157838486, 9.03016033957630277969788964747, 10.17934373182192740251309289094, 10.8376567867384747286210191523, 11.40976788557887954126202267729, 12.34638191978447748787679738213, 12.7020913667526691188896034788, 13.6073083814307335185832067223, 14.28982046974341561746005964223, 15.17043514221208492412985448504, 15.70529863108873281982200645349, 16.516425114397896323701830359408, 17.33399566748995686937296941610, 17.76166967780804906213761985695, 18.437431973668493093275603895858

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