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Properties

Label 2-2268-252.167-c0-0-7
Degree $2$
Conductor $2268$
Sign $0.573 - 0.819i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (0.965 + 1.67i)11-s + (−0.258 + 0.965i)14-s − 1.00·16-s + (−1.86 − 0.5i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.500 − 0.866i)28-s + (−1.22 + 0.707i)29-s + (0.707 − 0.707i)32-s − 1.73·37-s + (0.866 − 0.5i)43-s + (1.67 − 0.965i)44-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (0.965 + 1.67i)11-s + (−0.258 + 0.965i)14-s − 1.00·16-s + (−1.86 − 0.5i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.500 − 0.866i)28-s + (−1.22 + 0.707i)29-s + (0.707 − 0.707i)32-s − 1.73·37-s + (0.866 − 0.5i)43-s + (1.67 − 0.965i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.573 - 0.819i$
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: \((2,\ 2268,\ (\ :0),\ 0.573 - 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9704076644\)
\(L(\frac12)\) \(\approx\) \(0.9704076644\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.73T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 0.517iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 0.517T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
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

−9.093464765676589296825774786874, −8.704230045300601227377264857827, −7.56658046062529934061080699664, −7.15479298353403202191558312846, −6.57336421719641624607398163552, −5.31070338317774423776572101999, −4.76138114368692506796923807707, −3.87646643072053735821346212892, −2.11920904995409106692095585535, −1.30387920037482817024666121433, 1.03987831694922271318534620367, 2.08426943043975678766304158289, 3.24172750324410946793415488635, 3.91389786416576531500515034506, 5.08217087497700227930657947114, 5.97751302020872240907509843167, 6.93691323042475970510791705131, 7.87427737988619648268124647154, 8.494979607633772298621692127166, 9.050444836704433715711023602623

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