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Properties

Label 2-2268-252.83-c0-0-0
Degree $2$
Conductor $2268$
Sign $0.984 + 0.173i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 1.22i)11-s + (−0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + (1.36 + 0.366i)22-s + (0.707 + 1.22i)23-s + (0.5 − 0.866i)25-s + 28-s + (1.22 + 0.707i)29-s + (−0.965 − 0.258i)32-s + (1.73 + i)43-s − 1.41i·44-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 1.22i)11-s + (−0.258 + 0.965i)14-s + (0.500 − 0.866i)16-s + (1.36 + 0.366i)22-s + (0.707 + 1.22i)23-s + (0.5 − 0.866i)25-s + 28-s + (1.22 + 0.707i)29-s + (−0.965 − 0.258i)32-s + (1.73 + i)43-s − 1.41i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7556577325\)
\(L(\frac12)\) \(\approx\) \(0.7556577325\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
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 \( 1 + 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
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

−9.389113203581667454876287938939, −8.661459097897195002981114379659, −7.58727442115752064109251405576, −7.18165357392463559322561849019, −6.02169824965219995121429890551, −4.87631526355904665249142145189, −4.29537630258026084414631802717, −3.18965669183421912333595507609, −2.51475490380539186676254791718, −1.16839724627420651755693008545, 0.67354027204107503019520949603, 2.59675809418106401768756778408, 3.51641594548699916990092788828, 4.67411123228597668020552033252, 5.53216107973554730068331877537, 6.14086759766328125227105477119, 6.83695872473772458777834308322, 7.67807345791724768617330372774, 8.696968479514506655058476876738, 8.794566233571268982486317997598

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