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Properties

Label 2-2268-252.167-c0-0-8
Degree $2$
Conductor $2268$
Sign $0.0871 - 0.996i$
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 + (0.258 + 0.448i)11-s + (0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (−0.366 + 0.366i)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 + (0.965 + 0.258i)32-s + 1.73·37-s + (0.866 − 0.5i)43-s + (−0.448 − 0.258i)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 + (0.258 + 0.448i)11-s + (0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (−0.366 + 0.366i)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 + (0.965 + 0.258i)32-s + 1.73·37-s + (0.866 − 0.5i)43-s + (−0.448 − 0.258i)44-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.358031428\)
\(L(\frac12)\) \(\approx\) \(1.358031428\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 1.93iT - T^{2} \)
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 \( 1 + 1.93T + T^{2} \)
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

−9.233206853699239033630202267032, −8.446305655237215021886722305081, −7.57378401132469505552791869221, −7.33485432899765466560848176764, −6.21351656936421063402118767998, −5.55448875738906137436331039117, −4.52931081138681978805559026796, −4.13278411068795366262171829759, −2.88613039795436815491106179330, −1.32245215164252951105350974779, 1.07282001985636962221338768360, 2.28351530978771990606124622162, 3.01440143136858838978983986230, 4.30927130118260779264627462589, 4.71486802794343878513199806105, 5.79243373364524186894161891716, 6.40857204261009688343535081379, 7.75672708094817211412654537689, 8.586674576146837852350636662391, 8.881958754868903641289227612439

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