↑

Properties

Label 1-4031-4031.365-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.995 + 0.0914i$
Analytic cond. $18.7198$
Root an. cond. $18.7198$
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  + (−0.136 − 0.990i)2-s + (0.631 − 0.775i)3-s + (−0.962 + 0.269i)4-s + (−0.682 − 0.730i)5-s + (−0.854 − 0.519i)6-s + (0.854 − 0.519i)7-s + (0.398 + 0.917i)8-s + (−0.203 − 0.979i)9-s + (−0.631 + 0.775i)10-s + (0.816 + 0.576i)11-s + (−0.398 + 0.917i)12-s + (0.775 + 0.631i)13-s + (−0.631 − 0.775i)14-s + (−0.997 + 0.0682i)15-s + (0.854 − 0.519i)16-s + (−0.887 + 0.460i)17-s + ⋯
L(s)  = 1  + (−0.136 − 0.990i)2-s + (0.631 − 0.775i)3-s + (−0.962 + 0.269i)4-s + (−0.682 − 0.730i)5-s + (−0.854 − 0.519i)6-s + (0.854 − 0.519i)7-s + (0.398 + 0.917i)8-s + (−0.203 − 0.979i)9-s + (−0.631 + 0.775i)10-s + (0.816 + 0.576i)11-s + (−0.398 + 0.917i)12-s + (0.775 + 0.631i)13-s + (−0.631 − 0.775i)14-s + (−0.997 + 0.0682i)15-s + (0.854 − 0.519i)16-s + (−0.887 + 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.995 + 0.0914i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.995 + 0.0914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6397888785 + 0.02933174324i\)
\(L(\frac12)\) \(\approx\) \(0.6397888785 + 0.02933174324i\)
\(L(1)\) \(\approx\) \(0.6888053021 - 0.6043952822i\)
\(L(1)\) \(\approx\) \(0.6888053021 - 0.6043952822i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.136 - 0.990i)T \)
3 \( 1 + (0.631 - 0.775i)T \)
5 \( 1 + (-0.682 - 0.730i)T \)
7 \( 1 + (0.854 - 0.519i)T \)
11 \( 1 + (0.816 + 0.576i)T \)
13 \( 1 + (0.775 + 0.631i)T \)
17 \( 1 + (-0.887 + 0.460i)T \)
19 \( 1 + (-0.887 + 0.460i)T \)
23 \( 1 + (-0.854 + 0.519i)T \)
31 \( 1 + (-0.979 - 0.203i)T \)
37 \( 1 + (-0.997 - 0.0682i)T \)
41 \( 1 + (-0.942 - 0.334i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.730 + 0.682i)T \)
53 \( 1 + (0.990 - 0.136i)T \)
59 \( 1 + (-0.962 - 0.269i)T \)
61 \( 1 + (0.942 - 0.334i)T \)
67 \( 1 + (0.576 + 0.816i)T \)
71 \( 1 + (-0.682 - 0.730i)T \)
73 \( 1 + (0.398 + 0.917i)T \)
79 \( 1 + (-0.942 + 0.334i)T \)
83 \( 1 + (-0.576 + 0.816i)T \)
89 \( 1 + (-0.997 + 0.0682i)T \)
97 \( 1 - iT \)
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

−18.54603079561722192342902401632, −17.7181018180707946852856380860, −17.01393655194986010080277885426, −16.13511640467911257750474611232, −15.66078781207863119480423147966, −15.053371399318589804480921958888, −14.69208685245458327853632312974, −13.904373411064094058377001331013, −13.48952008662098735735555521679, −12.27905709063986030189418009570, −11.37601724031659103594836381568, −10.73579995075697039620349184295, −10.21207802258717764113062970776, −8.95221639696723373694765582223, −8.70581914681382438408500135252, −8.19183863999191321127840931671, −7.32428127619555536779219360174, −6.598587030659033978194637553632, −5.75326062224182702467697984543, −4.99250447998780187905651045320, −4.101158858872289830041861411799, −3.75597219834035644213039165867, −2.730038366079691871133253027484, −1.725002310499052427605959196299, −0.17436528462796258951691595690, 1.17690483978206095368377172440, 1.6385551514761343282599360223, 2.19647759988641584778408996872, 3.67326156018794093765455384290, 3.95309330264769174962877817461, 4.53493098516855347255209592708, 5.67588984336402741788026445734, 6.7744847276560322435120588031, 7.51367876245995679233697877035, 8.32061405171798844866727994442, 8.65855390608588553669051235712, 9.28413434128517300982524809015, 10.22597250781186663244404801788, 11.27742867730085384627961403499, 11.5266822059053187393204853685, 12.38745620792124682400142457650, 12.82457087694746420091309605525, 13.62093686094070751400628029901, 14.17539691223837208801498274274, 14.82230733762471943867151240713, 15.65007922107612687376088809303, 16.840669558099058492589426031014, 17.24695224131955865195251502025, 17.95551598874166945335949933835, 18.616856084858159883539762191159

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