Properties

Label 2-2268-63.31-c0-0-1
Degree $2$
Conductor $2268$
Sign $0.975 + 0.220i$
Analytic cond. $1.13187$
Root an. cond. $1.06389$
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  + (0.5 − 0.866i)7-s + (1.5 + 0.866i)13-s + (−1.5 + 0.866i)19-s + 25-s + (1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)67-s + (1.5 + 0.866i)73-s + (0.5 − 0.866i)79-s + (1.5 − 0.866i)91-s + 1.73i·103-s + (−0.5 + 0.866i)109-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)7-s + (1.5 + 0.866i)13-s + (−1.5 + 0.866i)19-s + 25-s + (1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)67-s + (1.5 + 0.866i)73-s + (0.5 − 0.866i)79-s + (1.5 − 0.866i)91-s + 1.73i·103-s + (−0.5 + 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.975 + 0.220i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.975 + 0.220i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.322829509\)
\(L(\frac12)\) \(\approx\) \(1.322829509\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.979441483544332929766902880955, −8.442678181910493699750935129733, −7.75667065823235830401726484025, −6.67215960365943871774579036999, −6.31147507001122015262764068857, −5.14550878103349304957407066952, −4.13632102021669576111453616616, −3.75313942168689720041166296671, −2.24045872319457536057613420701, −1.18152064925248947837004719798, 1.26990601036553926798128041157, 2.52660248627938250576905248945, 3.36504896752369554051610435511, 4.57492324675834121825223765147, 5.20732753230937849669996872415, 6.26784849691681676767382426068, 6.63852731943742718201619396407, 8.024350534442175993078598384048, 8.473137596244116041885483283870, 8.958941341863152503021898675991

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