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Properties

Label 1-2268-2268.67-r1-0-0
Degree $1$
Conductor $2268$
Sign $0.0141 + 0.999i$
Analytic cond. $243.730$
Root an. cond. $243.730$
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  + (0.893 + 0.448i)5-s + (0.0581 − 0.998i)11-s + (−0.993 − 0.116i)13-s + (−0.939 − 0.342i)17-s + (0.939 − 0.342i)19-s + (0.286 − 0.957i)23-s + (0.597 + 0.802i)25-s + (0.597 + 0.802i)29-s + (−0.973 + 0.230i)31-s + (−0.939 − 0.342i)37-s + (0.396 + 0.918i)41-s + (−0.893 + 0.448i)43-s + (−0.973 − 0.230i)47-s + (−0.5 + 0.866i)53-s + (0.5 − 0.866i)55-s + ⋯
L(s)  = 1  + (0.893 + 0.448i)5-s + (0.0581 − 0.998i)11-s + (−0.993 − 0.116i)13-s + (−0.939 − 0.342i)17-s + (0.939 − 0.342i)19-s + (0.286 − 0.957i)23-s + (0.597 + 0.802i)25-s + (0.597 + 0.802i)29-s + (−0.973 + 0.230i)31-s + (−0.939 − 0.342i)37-s + (0.396 + 0.918i)41-s + (−0.893 + 0.448i)43-s + (−0.973 − 0.230i)47-s + (−0.5 + 0.866i)53-s + (0.5 − 0.866i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0141 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0141 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.0141 + 0.999i$
Analytic conductor: \(243.730\)
Root analytic conductor: \(243.730\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2268,\ (1:\ ),\ 0.0141 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.059102639 + 1.074244310i\)
\(L(\frac12)\) \(\approx\) \(1.059102639 + 1.074244310i\)
\(L(1)\) \(\approx\) \(1.087798342 + 0.06513784049i\)
\(L(1)\) \(\approx\) \(1.087798342 + 0.06513784049i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (0.0581 - 0.998i)T \)
13 \( 1 + (-0.993 - 0.116i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.286 - 0.957i)T \)
29 \( 1 + (0.597 + 0.802i)T \)
31 \( 1 + (-0.973 + 0.230i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (0.396 + 0.918i)T \)
43 \( 1 + (-0.893 + 0.448i)T \)
47 \( 1 + (-0.973 - 0.230i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.893 - 0.448i)T \)
61 \( 1 + (-0.286 - 0.957i)T \)
67 \( 1 + (0.993 + 0.116i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.597 - 0.802i)T \)
83 \( 1 + (-0.396 + 0.918i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.0581 + 0.998i)T \)
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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

−19.52495691498350441622519705748, −18.42196205982397132726202451511, −17.697477805211305097275156914662, −17.306295493125364391484115868919, −16.59994249269522916523636103481, −15.62830316462869503505215013877, −15.03172169034293284150922699527, −14.13493199420863440720741160587, −13.557744369938420725869694990788, −12.73161650632629958208360566668, −12.180281417438849427593956100002, −11.32966732522229214240776851272, −10.28132334505109647635670751966, −9.691856494793691323657998004638, −9.20287289086320010417452156068, −8.22630777259223281947732725431, −7.28356746854903051186606457199, −6.679714910551051540813184862688, −5.63509486823749390526937340048, −5.018828852393018099705949977703, −4.28868182678472519925503393874, −3.14712120198215530990189412125, −2.05265531256454245157105140417, −1.637828292595999852991877448919, −0.27002596196692020102756376955, 0.85802558686667850115866986911, 1.9513069984576097783326163082, 2.81125315479021046980592548805, 3.40588058331763142823661415823, 4.842731247469133468490019631527, 5.241648809346003922520476773745, 6.363472627512865998475959960453, 6.80580634318700041184141693160, 7.73889515155460395630114090436, 8.770857512020480682102942081225, 9.34741425731807145392338500596, 10.13636345327588866594217816808, 10.91670800002423270332729071026, 11.46289445658737416429416287045, 12.568261672563992925492376675840, 13.1491393088102480476185399507, 14.17501670135912186791870567279, 14.26046770683881176732633113304, 15.33913032845096236628511510512, 16.14953413309764145962718728826, 16.85918937038191214906899711882, 17.55802482648385459621748808711, 18.28748628236454551008081510136, 18.73812228951228558062081516713, 19.86613265922301343016739590088

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