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Properties

Label 2-2268-63.40-c0-0-0
Degree $2$
Conductor $2268$
Sign $0.888 + 0.458i$
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)7-s + (1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s − 1.73i·31-s + (1 − 1.73i)37-s + (−0.5 − 0.866i)43-s + (−0.499 − 0.866i)49-s + 1.73i·61-s + 2·67-s + (−1.5 + 0.866i)73-s + 2·79-s + (−1.5 + 0.866i)97-s + (−0.5 − 0.866i)109-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)7-s + (1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s − 1.73i·31-s + (1 − 1.73i)37-s + (−0.5 − 0.866i)43-s + (−0.499 − 0.866i)49-s + 1.73i·61-s + 2·67-s + (−1.5 + 0.866i)73-s + 2·79-s + (−1.5 + 0.866i)97-s + (−0.5 − 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.888 + 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.284538936\)
\(L(\frac12)\) \(\approx\) \(1.284538936\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
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 \( 1 - T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.73iT - T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 - 2T + T^{2} \)
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

−9.323767556772386123464650195844, −8.183390758650585172172434113875, −7.59959335662148466210898760268, −7.06708291030674480868014192094, −5.86504623954550945699731926177, −5.29885318682443581622708509130, −4.14460132046167396445146082816, −3.60346386295415505963506343384, −2.26393279897665724332223860809, −1.06753734976265999161598554399, 1.34889658328521254773296566407, 2.57558457321869197982958817010, 3.35829728128863022803359499921, 4.73595378358401665628286954695, 5.14561318615506452647437531644, 6.17088919429311617222424859917, 6.88783576059875584445743675222, 7.926850084592311896253901883400, 8.390185368178959278672296847582, 9.332015813663208421338619012778

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