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Properties

Label 2-2268-63.52-c0-0-0
Degree $2$
Conductor $2268$
Sign $-0.841 - 0.540i$
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 + (−1.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + 1.73i·31-s + (−0.5 − 0.866i)37-s + (−0.5 + 0.866i)43-s + 49-s − 67-s + (1.5 + 0.866i)73-s − 79-s + (1.5 − 0.866i)91-s + (−1.5 − 0.866i)103-s + (−0.5 + 0.866i)109-s + ⋯
L(s)  = 1  − 7-s + (−1.5 + 0.866i)13-s + (−1.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + 1.73i·31-s + (−0.5 − 0.866i)37-s + (−0.5 + 0.866i)43-s + 49-s − 67-s + (1.5 + 0.866i)73-s − 79-s + (1.5 − 0.866i)91-s + (−1.5 − 0.866i)103-s + (−0.5 + 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.841 - 0.540i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1837, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ -0.841 - 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3226808911\)
\(L(\frac12)\) \(\approx\) \(0.3226808911\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
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 \( 1 - T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + T + T^{2} \)
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

−9.574204820740111496243966049823, −8.819159151122178194926460403699, −8.024770078571018137125466061662, −7.00081280337659589689183162227, −6.58279203004158094002444953168, −5.67479181888446398596094503121, −4.64136921014109182481151666043, −3.90316651102538969112221399413, −2.80711424125016722309785520725, −1.89191983440623025157436022048, 0.19936798300695318836333084695, 2.17775552206390540664676417279, 2.96457243555450345273859150733, 3.98683951580907400592450114206, 4.92982020567103193352330236177, 5.77702885276373498323598161659, 6.61833868672080014786231068553, 7.30208972662518895185062778617, 8.075151984491076032076500333442, 9.018151897168569056718004338847

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