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Properties

Label 2-2268-63.4-c1-0-17
Degree $2$
Conductor $2268$
Sign $0.999 - 0.00822i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.343·5-s + (2.14 + 1.55i)7-s + 4.91·11-s + (0.974 − 1.68i)13-s + (2.07 − 3.59i)17-s + (0.202 + 0.350i)19-s + 2.75·23-s − 4.88·25-s + (1.63 + 2.83i)29-s + (1.64 + 2.84i)31-s + (0.736 + 0.533i)35-s + (−3.38 − 5.87i)37-s + (−2.81 + 4.87i)41-s + (−4.96 − 8.59i)43-s + (6.60 − 11.4i)47-s + ⋯
L(s)  = 1  + 0.153·5-s + (0.810 + 0.586i)7-s + 1.48·11-s + (0.270 − 0.467i)13-s + (0.502 − 0.871i)17-s + (0.0464 + 0.0803i)19-s + 0.575·23-s − 0.976·25-s + (0.304 + 0.527i)29-s + (0.295 + 0.511i)31-s + (0.124 + 0.0901i)35-s + (−0.557 − 0.965i)37-s + (−0.439 + 0.760i)41-s + (−0.756 − 1.31i)43-s + (0.962 − 1.66i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.999 - 0.00822i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.999 - 0.00822i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.331368808\)
\(L(\frac12)\) \(\approx\) \(2.331368808\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 - 1.55i)T \)
good5 \( 1 - 0.343T + 5T^{2} \)
11 \( 1 - 4.91T + 11T^{2} \)
13 \( 1 + (-0.974 + 1.68i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.202 - 0.350i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.75T + 23T^{2} \)
29 \( 1 + (-1.63 - 2.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.64 - 2.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.38 + 5.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.81 - 4.87i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.96 + 8.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.60 + 11.4i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.38 + 2.39i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.22 - 7.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.37 - 9.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.21 - 7.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.08T + 71T^{2} \)
73 \( 1 + (-4.58 + 7.93i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.24 - 3.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.62 - 8.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.54 - 6.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.31 + 2.27i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.850101280158688940309017789866, −8.505731136006064067772316524714, −7.39687977570800388800098954224, −6.78855335936597472755560721983, −5.72193397039326467153738468035, −5.20589176025265334383222400141, −4.14723030545703897747140726272, −3.26213685880434849726955723024, −2.06907201367401830813800157370, −1.05729006709867879411382822688, 1.12945739400494823974472594182, 1.90109390078348733104043078930, 3.43168274693255524804599065871, 4.15715992831239278362745897297, 4.89585171186591774614662688618, 6.08391257684383702190199822017, 6.57089861126565383475015585665, 7.56911235793509865802239602610, 8.215352115229133053390656403416, 9.027709810575488066024544691275

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