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Properties

Label 2-2268-21.17-c1-0-21
Degree $2$
Conductor $2268$
Sign $0.758 + 0.651i$
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  + (0.566 + 0.981i)5-s + (2.58 − 0.553i)7-s + (−2.30 − 1.32i)11-s − 1.84i·13-s + (0.267 − 0.463i)17-s + (2.57 − 1.48i)19-s + (0.839 − 0.484i)23-s + (1.85 − 3.21i)25-s − 3.08i·29-s + (0.682 + 0.394i)31-s + (2.00 + 2.22i)35-s + (−1.73 − 3.00i)37-s − 4.45·41-s + 1.69·43-s + (−5.14 − 8.90i)47-s + ⋯
L(s)  = 1  + (0.253 + 0.438i)5-s + (0.977 − 0.209i)7-s + (−0.693 − 0.400i)11-s − 0.510i·13-s + (0.0648 − 0.112i)17-s + (0.590 − 0.340i)19-s + (0.175 − 0.101i)23-s + (0.371 − 0.643i)25-s − 0.572i·29-s + (0.122 + 0.0707i)31-s + (0.339 + 0.376i)35-s + (−0.284 − 0.493i)37-s − 0.695·41-s + 0.259·43-s + (−0.750 − 1.29i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.758 + 0.651i$
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: \((2,\ 2268,\ (\ :1/2),\ 0.758 + 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.970823722\)
\(L(\frac12)\) \(\approx\) \(1.970823722\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.58 + 0.553i)T \)
good5 \( 1 + (-0.566 - 0.981i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.30 + 1.32i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.84iT - 13T^{2} \)
17 \( 1 + (-0.267 + 0.463i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.57 + 1.48i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.839 + 0.484i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.08iT - 29T^{2} \)
31 \( 1 + (-0.682 - 0.394i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.73 + 3.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.45T + 41T^{2} \)
43 \( 1 - 1.69T + 43T^{2} \)
47 \( 1 + (5.14 + 8.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.9 - 6.87i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.64 + 6.31i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.74 + 1.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.20 + 2.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + (-7.05 - 4.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.58 + 4.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 + (-6.52 - 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.66iT - 97T^{2} \)
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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

−8.726056951864375275357580740773, −8.211088206561719589693369450891, −7.41301550253206268284727503913, −6.71610572113493304418228702554, −5.60224289093587492976863919230, −5.12054310594016319028338375735, −4.07982738488754753301472297784, −2.99523350549339433733048896623, −2.14587915941123489014865893527, −0.74792116943843332454419463876, 1.24758905037503440078092511594, 2.13045954895524830700498335387, 3.30720416853483263679488227248, 4.50385229815254583671314865150, 5.11224991977729826616639060508, 5.76203923991187577404776667304, 6.92200281669192595920609981117, 7.61073125296553528734614637908, 8.396264420281164063407736746014, 9.006117493043951657910457675641

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