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Properties

Label 2-2268-63.16-c1-0-29
Degree $2$
Conductor $2268$
Sign $-0.641 + 0.767i$
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  + 1.90·5-s + (−2.64 − 0.163i)7-s + 2.82·11-s + (−2.41 − 4.17i)13-s + (−2.14 − 3.70i)17-s + (2.37 − 4.11i)19-s − 2.46·23-s − 1.37·25-s + (−4.32 + 7.49i)29-s + (−1.68 + 2.91i)31-s + (−5.02 − 0.310i)35-s + (−2.59 + 4.48i)37-s + (−4.10 − 7.10i)41-s + (−3.36 + 5.82i)43-s + (−0.864 − 1.49i)47-s + ⋯
L(s)  = 1  + 0.851·5-s + (−0.998 − 0.0616i)7-s + 0.852·11-s + (−0.669 − 1.15i)13-s + (−0.519 − 0.899i)17-s + (0.544 − 0.943i)19-s − 0.514·23-s − 0.275·25-s + (−0.803 + 1.39i)29-s + (−0.302 + 0.524i)31-s + (−0.849 − 0.0524i)35-s + (−0.426 + 0.738i)37-s + (−0.640 − 1.10i)41-s + (−0.513 + 0.888i)43-s + (−0.126 − 0.218i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.641 + 0.767i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.641 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9723190088\)
\(L(\frac12)\) \(\approx\) \(0.9723190088\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.163i)T \)
good5 \( 1 - 1.90T + 5T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + (2.41 + 4.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.14 + 3.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.37 + 4.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.46T + 23T^{2} \)
29 \( 1 + (4.32 - 7.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.68 - 2.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.59 - 4.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.10 + 7.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.36 - 5.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.864 + 1.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.80 + 10.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.73 + 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.62 + 4.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.35 + 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.39T + 71T^{2} \)
73 \( 1 + (-4.23 - 7.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.97 - 3.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.72 + 8.18i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.91 - 8.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.60 - 2.78i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.015434373218387421457082458963, −7.969206895097090107954853508756, −6.84617458443832269382333315240, −6.67568181173955084707773638993, −5.46176456762967182313565139827, −5.01919146650536885735870208917, −3.60192095558514537246364466014, −2.93255701287329089807610305851, −1.81153302748960951270946843841, −0.30806769887984025941281036622, 1.61560065563005053085442857019, 2.38307150792236887834903185601, 3.72669291621433518608319042742, 4.27273225738102731416520736617, 5.68752569388346702977579744962, 6.11101315265118574913916337643, 6.82881681296656166151268685342, 7.67068406773877140818887379218, 8.764447039741412217412391173022, 9.463676879113321240072752654791

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