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Properties

Label 2-2268-252.167-c0-0-0
Degree $2$
Conductor $2268$
Sign $-0.342 + 0.939i$
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)2-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 0.999·8-s − 0.999·10-s + (−0.5 − 0.866i)11-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 2·17-s − 19-s + (−0.499 − 0.866i)20-s + (0.499 − 0.866i)22-s + (−0.5 + 0.866i)23-s + 0.999·28-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 0.999·8-s − 0.999·10-s + (−0.5 − 0.866i)11-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 2·17-s − 19-s + (−0.499 − 0.866i)20-s + (0.499 − 0.866i)22-s + (−0.5 + 0.866i)23-s + 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1056319361\)
\(L(\frac12)\) \(\approx\) \(0.1056319361\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.611528101627182073541016726324, −8.709925750952979554698179846700, −8.021798460445972395354784225105, −7.27914891303220673270311880185, −6.58636044361901304732319538454, −6.16742380293145663754479694015, −4.95591686772871293700804988672, −4.05560100228629968758842280749, −3.46958458099464259555162702871, −2.48767257372128107664000793652, 0.05589247536055822244862706505, 1.95770923021089394263121005602, 2.56828873694785192864103702697, 3.86801365133596540582806283427, 4.63512055522345209785324507288, 5.10990659886710935849476976855, 6.26219586184096792929289563233, 6.86273479009930777110412205152, 8.407984780705646861222181187635, 8.683853055733140379931782060656

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