Properties

Label 2-2268-63.61-c0-0-0
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  − 7-s + (1.5 + 0.866i)19-s + 25-s + (1.5 + 0.866i)31-s + (1 − 1.73i)37-s + (−0.5 + 0.866i)43-s + 49-s + (1.5 − 0.866i)61-s + (−1 + 1.73i)67-s + (−1.5 + 0.866i)73-s + (−1 − 1.73i)79-s + (1.5 + 0.866i)97-s + (−0.5 − 0.866i)109-s + ⋯
L(s)  = 1  − 7-s + (1.5 + 0.866i)19-s + 25-s + (1.5 + 0.866i)31-s + (1 − 1.73i)37-s + (−0.5 + 0.866i)43-s + 49-s + (1.5 − 0.866i)61-s + (−1 + 1.73i)67-s + (−1.5 + 0.866i)73-s + (−1 − 1.73i)79-s + (1.5 + 0.866i)97-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: $\chi_{2268} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.975 - 0.220i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.101905686\)
\(L(\frac12)\) \(\approx\) \(1.101905686\)
\(L(1)\) 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 \( 1 + T \)
good5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.294327477107004576495528176382, −8.537055831210476786177897564355, −7.61641572034219727366089564199, −6.93114040587752945451741525712, −6.11255882673049950569124607722, −5.39734720768433834784366247044, −4.36095524852641853657839353913, −3.36974865705466415476703235333, −2.68060789622532968871999362027, −1.12886443883065970027101895307, 0.980121264161258131646835879584, 2.65996633537288379440909000185, 3.22835535128256897174525249439, 4.38289635081508370721195329329, 5.21457309656645287248648761666, 6.17625995762833166141827655367, 6.81422556148650222328618502872, 7.56422410661195451642873117107, 8.486742116533799096673114874733, 9.261796085787225578427612449624

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