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Properties

Label 2-2268-63.34-c0-0-2
Degree $2$
Conductor $2268$
Sign $0.766 + 0.642i$
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)13-s − 1.73i·19-s + (−0.5 + 0.866i)25-s + 37-s + (1 − 1.73i)43-s + 49-s + (1.5 + 0.866i)61-s + (0.5 + 0.866i)67-s − 1.73i·73-s + (0.5 − 0.866i)79-s + (−1.5 + 0.866i)91-s + (−1.5 − 0.866i)97-s + (−1.5 + 0.866i)103-s − 2·109-s + ⋯
L(s)  = 1  − 7-s + (1.5 − 0.866i)13-s − 1.73i·19-s + (−0.5 + 0.866i)25-s + 37-s + (1 − 1.73i)43-s + 49-s + (1.5 + 0.866i)61-s + (0.5 + 0.866i)67-s − 1.73i·73-s + (0.5 − 0.866i)79-s + (−1.5 + 0.866i)91-s + (−1.5 − 0.866i)97-s + (−1.5 + 0.866i)103-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.083272840\)
\(L(\frac12)\) \(\approx\) \(1.083272840\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
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

−9.086904130374747699671617419812, −8.515269951709445671315061169925, −7.49380280766011637266041688068, −6.79195093489357723926246348362, −5.98068878866284274193485710456, −5.34308692405459366573297979153, −4.10244163824102482682974948997, −3.36232265057665199431457959435, −2.47277485950077349043433255441, −0.851976924933049849242585471126, 1.32527662649462541694657461225, 2.59468350929736852745878501879, 3.75565811128662612243217467035, 4.14613535989302176557514583390, 5.60207432715125044375550320083, 6.23624959591087673103876090960, 6.73913834344124636862697925753, 7.942301066710654382239133360488, 8.424365657196634871642871830394, 9.490457717169550062306459628965

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