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Properties

Label 1-1155-1155.482-r1-0-0
Degree $1$
Conductor $1155$
Sign $0.564 - 0.825i$
Analytic cond. $124.121$
Root an. cond. $124.121$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.951 + 0.309i)8-s + (0.587 − 0.809i)13-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.809i)17-s + (0.309 − 0.951i)19-s + i·23-s + (0.309 + 0.951i)26-s + (0.309 + 0.951i)29-s + (0.809 + 0.587i)31-s − i·32-s + 34-s + (−0.951 + 0.309i)37-s + (0.587 + 0.809i)38-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.951 + 0.309i)8-s + (0.587 − 0.809i)13-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.809i)17-s + (0.309 − 0.951i)19-s + i·23-s + (0.309 + 0.951i)26-s + (0.309 + 0.951i)29-s + (0.809 + 0.587i)31-s − i·32-s + 34-s + (−0.951 + 0.309i)37-s + (0.587 + 0.809i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(124.121\)
Root analytic conductor: \(124.121\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1155,\ (1:\ ),\ 0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.007443461 - 0.5318467816i\)
\(L(\frac12)\) \(\approx\) \(1.007443461 - 0.5318467816i\)
\(L(1)\) \(\approx\) \(0.7786131680 + 0.1160424850i\)
\(L(1)\) \(\approx\) \(0.7786131680 + 0.1160424850i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.07339907714229476288864580289, −20.496116458939593688448498690114, −19.57046840080376578835268391173, −18.94372803908844135948142634044, −18.29544189924277141000845811245, −17.4691997988269292484656825050, −16.702386823548870119162977989220, −16.06656349806823689326118870797, −15.00916082657250404963295227759, −13.9426232476661439082345334861, −13.321316629206919223158742378884, −12.36257645686262681174259381102, −11.75345010814020466604223102007, −10.87131013879294730811872928871, −10.21405219531266460481230375067, −9.34063722070707491728226871965, −8.52370334184107619141233700506, −7.905985992157419520478007356950, −6.78140620453737379312160723890, −5.933622465663923565820812634272, −4.44336842675269988674663147405, −3.9656623204986260068441742168, −2.789967953904592334599704697883, −1.89140909584298367156732410417, −0.94724798771068707755747022523, 0.351426377100543308706163796107, 1.26907439278751516053219079630, 2.561495776209238799039491741210, 3.75855903860210379216024222825, 5.031034673711022897387677751130, 5.48521517335056759273424980046, 6.70870313946377061432132108073, 7.182003784635139119094632751488, 8.2419476329183095423246756858, 8.8939072918041547671332953022, 9.68498492559735158392910705365, 10.613329342125648799839713788583, 11.24616233569042993328373069483, 12.37909385628883428352639523140, 13.5845042677466013041788845498, 13.841853653922884098074449815688, 15.04126760410472786201117645133, 15.73996778413401895533401838554, 16.078481369847542205803012303078, 17.43919701464949360858359970470, 17.6133240700657595294668509228, 18.50849141170467266595876294139, 19.319820680806448393393902108434, 20.07022302079401064485768734701, 20.74066433287927540231587053130

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