Properties

Degree $2$
Conductor $2268$
Sign $-0.466 + 0.884i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.5i)5-s + (−1.62 + 2.09i)7-s + (3.67 − 2.12i)11-s + (−2.12 − 1.22i)13-s − 1.73·17-s − 2.44i·19-s + (−7.34 − 4.24i)23-s + (1 + 1.73i)25-s + (3.67 − 2.12i)29-s + (−6.36 − 3.67i)31-s + (1.73 + 4.24i)35-s + 37-s + (−0.866 + 1.5i)41-s + (−3.5 − 6.06i)43-s + (6.06 + 10.5i)47-s + ⋯
L(s)  = 1  + (0.387 − 0.670i)5-s + (−0.612 + 0.790i)7-s + (1.10 − 0.639i)11-s + (−0.588 − 0.339i)13-s − 0.420·17-s − 0.561i·19-s + (−1.53 − 0.884i)23-s + (0.200 + 0.346i)25-s + (0.682 − 0.393i)29-s + (−1.14 − 0.659i)31-s + (0.292 + 0.717i)35-s + 0.164·37-s + (−0.135 + 0.234i)41-s + (−0.533 − 0.924i)43-s + (0.884 + 1.53i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.466 + 0.884i$
Motivic weight: \(1\)
Character: $\chi_{2268} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.466 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.130003402\)
\(L(\frac12)\) \(\approx\) \(1.130003402\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 2.09i)T \)
good5 \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 + 2.44iT - 19T^{2} \)
23 \( 1 + (7.34 + 4.24i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.67 + 2.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.36 + 3.67i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (0.866 - 1.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.06 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-4.33 + 7.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.12 - 1.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.06 + 10.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (-2.12 + 1.22i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.849950104536672211182345576014, −8.248854209300262648265848710330, −7.13015104713568688117955416095, −6.23908994311134362331404556227, −5.77383138078713442693425159169, −4.80127899311515555304139166859, −3.90137551496666585400934476229, −2.81621323096548305166420930373, −1.84333491368649619061893591297, −0.37670627136664649667685616976, 1.46078960532197698852893184609, 2.49359136042701177376134405642, 3.70327274258445749065813927654, 4.19595520815506785794525403622, 5.41069507527942382300581682412, 6.41996656462520248859563373881, 6.86695545534883471915257983278, 7.49360201248938750367203733486, 8.571358187140873774766008210806, 9.486869579936277668472782189174

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