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Properties

Label 2-2268-21.17-c1-0-26
Degree $2$
Conductor $2268$
Sign $-0.775 + 0.630i$
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  + (−0.896 − 1.55i)5-s + (−0.0850 − 2.64i)7-s + (1.55 + 0.899i)11-s − 3.65i·13-s + (0.266 − 0.461i)17-s + (−2.21 + 1.27i)19-s + (7.27 − 4.20i)23-s + (0.891 − 1.54i)25-s + 6.87i·29-s + (−2.55 − 1.47i)31-s + (−4.03 + 2.50i)35-s + (−0.449 − 0.778i)37-s + 1.70·41-s − 10.3·43-s + (−3.11 − 5.39i)47-s + ⋯
L(s)  = 1  + (−0.401 − 0.694i)5-s + (−0.0321 − 0.999i)7-s + (0.469 + 0.271i)11-s − 1.01i·13-s + (0.0645 − 0.111i)17-s + (−0.507 + 0.292i)19-s + (1.51 − 0.876i)23-s + (0.178 − 0.308i)25-s + 1.27i·29-s + (−0.458 − 0.264i)31-s + (−0.681 + 0.423i)35-s + (−0.0739 − 0.128i)37-s + 0.266·41-s − 1.57·43-s + (−0.453 − 0.786i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.775 + 0.630i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.775 + 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.198950913\)
\(L(\frac12)\) \(\approx\) \(1.198950913\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0850 + 2.64i)T \)
good5 \( 1 + (0.896 + 1.55i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.55 - 0.899i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.65iT - 13T^{2} \)
17 \( 1 + (-0.266 + 0.461i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.21 - 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.27 + 4.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.87iT - 29T^{2} \)
31 \( 1 + (2.55 + 1.47i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.449 + 0.778i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.70T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + (3.11 + 5.39i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.153 + 0.0885i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.66 + 9.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.00 - 1.15i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.178 + 0.309i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.97iT - 71T^{2} \)
73 \( 1 + (11.0 + 6.35i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.29 - 7.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + (-6.52 - 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.2iT - 97T^{2} \)
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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

−8.574973675478575112571445124420, −8.093568080540987072773260523173, −7.09208646920210816655694609140, −6.62707521030901549877392049759, −5.33944906122968553771608534117, −4.71394796538575235376646215384, −3.86616932930905632128800202128, −3.00081280462731226759720593061, −1.45914069551304070133849777421, −0.43034087330538702558169024424, 1.55955575245299516106439876623, 2.69978637298409586174037167650, 3.48168867423466630881151222958, 4.48960108264909278730306575956, 5.42793764138194537368995927704, 6.33936515545174053648865602307, 6.91627535309057729338240926638, 7.72800698380618632346928964090, 8.764977650053424939184060822528, 9.125981472864610887424861146928

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