Properties

Label 1-1148-1148.467-r0-0-0
Degree $1$
Conductor $1148$
Sign $0.120 + 0.992i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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)3-s + (−0.669 + 0.743i)5-s + (−0.5 − 0.866i)9-s + (−0.669 − 0.743i)11-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)15-s + (−0.669 − 0.743i)17-s + (0.913 + 0.406i)19-s + (0.104 − 0.994i)23-s + (−0.104 − 0.994i)25-s + 27-s + (0.309 + 0.951i)29-s + (0.669 + 0.743i)31-s + (0.978 − 0.207i)33-s + (0.669 − 0.743i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.669 + 0.743i)5-s + (−0.5 − 0.866i)9-s + (−0.669 − 0.743i)11-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)15-s + (−0.669 − 0.743i)17-s + (0.913 + 0.406i)19-s + (0.104 − 0.994i)23-s + (−0.104 − 0.994i)25-s + 27-s + (0.309 + 0.951i)29-s + (0.669 + 0.743i)31-s + (0.978 − 0.207i)33-s + (0.669 − 0.743i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.120 + 0.992i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 0.120 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7199412988 + 0.6379501676i\)
\(L(\frac12)\) \(\approx\) \(0.7199412988 + 0.6379501676i\)
\(L(1)\) \(\approx\) \(0.7489339054 + 0.2932860549i\)
\(L(1)\) \(\approx\) \(0.7489339054 + 0.2932860549i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.669 + 0.743i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (0.978 + 0.207i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.913 - 0.406i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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

−20.92183661456018895238601317152, −20.25022933274990833658915651709, −19.5764291032443235490999069604, −18.82155578037810168768579037304, −17.90454911568785197257107224504, −17.439156695696029420829227242893, −16.56558451790582842257619848840, −15.56425564856249355966927413551, −15.28527194377285213863252702790, −13.65655303948121500496433992970, −13.25487407342818730273798233003, −12.5296533358872281464677228970, −11.71992697852957631869646834275, −11.13070922306478802610162542281, −10.11981869020835482491620767870, −9.01277308239534415802609689481, −7.98254527220561294134380068116, −7.70760376096357258618404704201, −6.60207654496512615120620894209, −5.662774210044929990382655124448, −4.92520757783257063608637255248, −3.9557140780911219157990532647, −2.70879983228180013125814540460, −1.58938350413911584833539964616, −0.61638298894318211647845738118, 0.833822333827061808545446480357, 2.68485688533096677626951221781, 3.35676038299048515988039018401, 4.27087059415754262984439266523, 5.09368626050261880813077959585, 6.15585030554817407230488121658, 6.79597753111837883940860946515, 7.93835832047134488067506482659, 8.77843921400232054522010166469, 9.65790969868399994724630580975, 10.734172998334787422044213891687, 11.029164631964837182535222748127, 11.77178964619689602704616679522, 12.71641011116597766287715447525, 14.03487834112903456593018310732, 14.38634779105157920519029175948, 15.5995626390167082500494128952, 16.0125411625338295303197765146, 16.46197854731950786024563003220, 17.81634728298168804492505913121, 18.28593819521171660602022010173, 19.06698612502341575969173091616, 20.07779377764962771126915033899, 20.80283660784058139279946551804, 21.51518478742727313207878200807

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