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Properties

Label 2-2268-21.17-c1-0-27
Degree $2$
Conductor $2268$
Sign $-0.998 - 0.0484i$
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  + (−0.349 − 0.605i)5-s + (−2.02 − 1.70i)7-s + (0.229 + 0.132i)11-s + 1.31i·13-s + (1.86 − 3.22i)17-s + (−0.382 + 0.220i)19-s + (−4.29 + 2.48i)23-s + (2.25 − 3.90i)25-s − 0.315i·29-s + (4.85 + 2.80i)31-s + (−0.322 + 1.82i)35-s + (−0.351 − 0.608i)37-s − 10.7·41-s − 7.46·43-s + (−3.50 − 6.06i)47-s + ⋯
L(s)  = 1  + (−0.156 − 0.270i)5-s + (−0.765 − 0.643i)7-s + (0.0692 + 0.0399i)11-s + 0.364i·13-s + (0.452 − 0.783i)17-s + (−0.0877 + 0.0506i)19-s + (−0.896 + 0.517i)23-s + (0.451 − 0.781i)25-s − 0.0585i·29-s + (0.872 + 0.503i)31-s + (−0.0545 + 0.308i)35-s + (−0.0577 − 0.0999i)37-s − 1.68·41-s − 1.13·43-s + (−0.510 − 0.884i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.998 - 0.0484i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.998 - 0.0484i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2770495087\)
\(L(\frac12)\) \(\approx\) \(0.2770495087\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.02 + 1.70i)T \)
good5 \( 1 + (0.349 + 0.605i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.229 - 0.132i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.31iT - 13T^{2} \)
17 \( 1 + (-1.86 + 3.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.382 - 0.220i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.29 - 2.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.315iT - 29T^{2} \)
31 \( 1 + (-4.85 - 2.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.351 + 0.608i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 7.46T + 43T^{2} \)
47 \( 1 + (3.50 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.51 + 4.91i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.73 - 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.89 + 2.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.97 + 5.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 + (6.66 + 3.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.698 + 1.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.44T + 83T^{2} \)
89 \( 1 + (5.59 + 9.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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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

−8.561062963468646477243181239238, −7.928037408152156253608662848618, −6.92328111557457490109760999533, −6.51684752569964646694671127126, −5.40612610876681326022841785264, −4.56029033904656325496639736596, −3.69342920839541705224518598790, −2.84914232623151886057503416059, −1.45655642677357254312987824763, −0.094193626790269842709552551621, 1.65076353856956981997215056070, 2.89336264234054078179346644177, 3.51079771863332576077975706245, 4.62462427740889495838357186070, 5.61874427509598255526483121131, 6.30934044870222824629979648696, 6.95409313638861955367100380979, 8.081563161601923267526696539833, 8.477124679238609363935821944600, 9.556050121763534863942876667479

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