↑

Properties

Label 1-66e2-4356.1127-r0-0-0
Degree $1$
Conductor $4356$
Sign $-0.964 + 0.263i$
Analytic cond. $20.2291$
Root an. cond. $20.2291$
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  + (−0.548 + 0.836i)5-s + (−0.997 − 0.0760i)7-s + (−0.969 + 0.244i)13-s + (0.870 − 0.491i)17-s + (0.736 − 0.676i)19-s + (0.235 − 0.971i)23-s + (−0.398 − 0.917i)25-s + (0.595 + 0.803i)29-s + (−0.999 − 0.0380i)31-s + (0.610 − 0.791i)35-s + (0.897 + 0.441i)37-s + (−0.797 + 0.603i)41-s + (−0.0475 + 0.998i)43-s + (−0.999 − 0.0190i)47-s + (0.988 + 0.151i)49-s + ⋯
L(s)  = 1  + (−0.548 + 0.836i)5-s + (−0.997 − 0.0760i)7-s + (−0.969 + 0.244i)13-s + (0.870 − 0.491i)17-s + (0.736 − 0.676i)19-s + (0.235 − 0.971i)23-s + (−0.398 − 0.917i)25-s + (0.595 + 0.803i)29-s + (−0.999 − 0.0380i)31-s + (0.610 − 0.791i)35-s + (0.897 + 0.441i)37-s + (−0.797 + 0.603i)41-s + (−0.0475 + 0.998i)43-s + (−0.999 − 0.0190i)47-s + (0.988 + 0.151i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4356\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-0.964 + 0.263i$
Analytic conductor: \(20.2291\)
Root analytic conductor: \(20.2291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4356} (1127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4356,\ (0:\ ),\ -0.964 + 0.263i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04669308318 + 0.3486440231i\)
\(L(\frac12)\) \(\approx\) \(0.04669308318 + 0.3486440231i\)
\(L(1)\) \(\approx\) \(0.7348536407 + 0.1163203121i\)
\(L(1)\) \(\approx\) \(0.7348536407 + 0.1163203121i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (-0.548 + 0.836i)T \)
7 \( 1 + (-0.997 - 0.0760i)T \)
13 \( 1 + (-0.969 + 0.244i)T \)
17 \( 1 + (0.870 - 0.491i)T \)
19 \( 1 + (0.736 - 0.676i)T \)
23 \( 1 + (0.235 - 0.971i)T \)
29 \( 1 + (0.595 + 0.803i)T \)
31 \( 1 + (-0.999 - 0.0380i)T \)
37 \( 1 + (0.897 + 0.441i)T \)
41 \( 1 + (-0.797 + 0.603i)T \)
43 \( 1 + (-0.0475 + 0.998i)T \)
47 \( 1 + (-0.999 - 0.0190i)T \)
53 \( 1 + (-0.941 + 0.336i)T \)
59 \( 1 + (0.123 - 0.992i)T \)
61 \( 1 + (0.820 - 0.572i)T \)
67 \( 1 + (0.327 + 0.945i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (-0.0285 + 0.999i)T \)
79 \( 1 + (-0.988 + 0.151i)T \)
83 \( 1 + (0.380 - 0.924i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (0.548 + 0.836i)T \)
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

−18.01635576462311266375459328502, −17.02886820386495105307305382197, −16.75859110764881093073753750403, −15.984711824796911882904317163, −15.46128720482984612199452963708, −14.72112174807616934470929478288, −13.90331550065489214817261426047, −13.05947533918720085183568619587, −12.586446114638262966561727948252, −11.99681286906790405424090219292, −11.40485027901078439812949351743, −10.207485943083630639799998800199, −9.79630966118405352734544018470, −9.13715924477552932042493916198, −8.28908832529206768625388323523, −7.57834390054838333881731460029, −7.0707328314844455023077484766, −5.87898016294735681369159262940, −5.47121309677514701382021813117, −4.60141428222034425440408463452, −3.636749897192504468812172169759, −3.263370940268927277124122139103, −2.104361210447772929148853867957, −1.12824993484987460354406538096, −0.11957045545442557227737775560, 0.98414527670056515634659650995, 2.361819269498176947923640391827, 3.03965775700735231236772351627, 3.457734942816884591374210819433, 4.56380040405602947636578090594, 5.19386515263516242553965705607, 6.33639431138647883827151971121, 6.79080246094591730000989976150, 7.43307959338478522049825242835, 8.10649529623861452774119320938, 9.164423766241457211184476361702, 9.81061830349027977718779046319, 10.27581215149727302815136876318, 11.24364449967503844127172454289, 11.729143771595308042856020529567, 12.57653830108546479818284258203, 13.051331074476772959689643740402, 14.164003640102916122788662693389, 14.483904624015253094783669380869, 15.21859129792272757610694300595, 16.11300841398731784245015536323, 16.3632573645949520526246925323, 17.23801961586465627424718585383, 18.16085023925000326694447471444, 18.6524290676805391772125115905

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