↑

Properties

Label 1-5544-5544.2419-r0-0-0
Degree $1$
Conductor $5544$
Sign $-0.281 + 0.959i$
Analytic cond. $25.7462$
Root an. cond. $25.7462$
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  + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s − 31-s + (0.5 − 0.866i)37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s − 47-s + (0.5 + 0.866i)53-s + 59-s + 61-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s − 31-s + (0.5 − 0.866i)37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s − 47-s + (0.5 + 0.866i)53-s + 59-s + 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5544\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.281 + 0.959i$
Analytic conductor: \(25.7462\)
Root analytic conductor: \(25.7462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5544} (2419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5544,\ (0:\ ),\ -0.281 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.064024904 + 1.421506152i\)
\(L(\frac12)\) \(\approx\) \(1.064024904 + 1.421506152i\)
\(L(1)\) \(\approx\) \(1.097835833 + 0.3498709779i\)
\(L(1)\) \(\approx\) \(1.097835833 + 0.3498709779i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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

−17.73308845045474293614560958675, −16.85410226797092112109130596457, −16.41764021075243067360882403895, −15.94450282829251340602114014455, −14.76036123162374628444043776890, −14.558515398193335915622176920760, −13.58905802102138324568750676266, −12.96028308831896290175056514624, −12.479255538630393369897585383290, −11.81357989972453883404878069782, −10.992521292186797314163138913148, −10.08827547820180555627778853643, −9.71934275680390964822586021570, −8.94493230770714664218456961883, −8.24199786234874640484517013393, −7.59617218074933840316394557212, −6.81835770551203697760328196594, −5.8761125419032751577326849548, −5.24385244355512417902758848881, −4.852390432451686479165913433596, −3.78678932682995339645960456124, −3.01207748934398866338431430086, −2.17453069810179714259983986911, −1.276288247280980206308778514827, −0.49293494565115823359363137259, 1.06138317954553307946928303869, 2.04998886116203897183723179265, 2.55080857129332145509637657095, 3.54714231559755637516652407324, 4.095105015030757720982559595457, 5.21771238070687750476702052073, 5.73977568843033774668113576296, 6.55215298600543450650728077104, 7.231335368257478980503866865449, 7.69040921071291444442276416713, 8.76546752647966835177874325220, 9.60173161688305217315930666119, 9.770862243858836374417295394240, 11.011238901998521313939552475371, 11.11212875127857626814488038565, 12.02548190746538445262931475082, 12.8770837245767888105976971070, 13.46103482341840279710933932959, 14.196992978647048574287422059869, 14.71279742677417092067708540471, 15.26410582510350322131690039151, 16.112329654523325566272957455527, 16.82078971700606806625887930096, 17.51508680280349694788195940950, 17.921066465523877065962546240649

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