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Properties

Label 2-2268-252.83-c0-0-11
Degree $2$
Conductor $2268$
Sign $0.996 - 0.0871i$
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  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (0.965 − 1.67i)11-s + (−0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (1.36 − 1.36i)22-s + (0.707 + 1.22i)23-s + (0.5 − 0.866i)25-s + (−0.5 − 0.866i)28-s + (1.22 + 0.707i)29-s + (0.258 + 0.965i)32-s − 1.73·37-s + (−0.866 − 0.5i)43-s + (1.67 − 0.965i)44-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (0.965 − 1.67i)11-s + (−0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (1.36 − 1.36i)22-s + (0.707 + 1.22i)23-s + (0.5 − 0.866i)25-s + (−0.5 − 0.866i)28-s + (1.22 + 0.707i)29-s + (0.258 + 0.965i)32-s − 1.73·37-s + (−0.866 − 0.5i)43-s + (1.67 − 0.965i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.996 - 0.0871i$
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: \((2,\ 2268,\ (\ :0),\ 0.996 - 0.0871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.141591796\)
\(L(\frac12)\) \(\approx\) \(2.141591796\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
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

−8.941990447451365794792018377674, −8.531276389437033900506937774293, −7.35626057203883836739299039941, −6.72581306841658871676138480784, −6.12551543247300527703718737994, −5.35063748448071755361310606278, −4.30929973643031598907153571091, −3.38703637044892145761383098612, −3.03120966339614864431093644634, −1.30464371774011452369715669469, 1.55844927099671325418970651658, 2.58807203377252978256114020624, 3.44763913704678022493756135255, 4.45024606204535529815417843318, 5.02135961151817535200233376777, 6.11733956866806790981330344645, 6.80540282015095959401857991337, 7.17433497713015317158427177677, 8.540638266932095392567431516315, 9.378943711459767541135123132259

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