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Properties

Label 2-2268-63.25-c1-0-11
Degree $2$
Conductor $2268$
Sign $0.760 - 0.649i$
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  + (0.171 − 0.297i)5-s + (0.271 − 2.63i)7-s + (2.45 + 4.26i)11-s + (0.974 + 1.68i)13-s + (−2.07 + 3.59i)17-s + (0.202 + 0.350i)19-s + (1.37 − 2.38i)23-s + (2.44 + 4.22i)25-s + (−1.63 + 2.83i)29-s − 3.28·31-s + (−0.736 − 0.533i)35-s + (−3.38 − 5.87i)37-s + (2.81 + 4.87i)41-s + (−4.96 + 8.59i)43-s + 13.2·47-s + ⋯
L(s)  = 1  + (0.0768 − 0.133i)5-s + (0.102 − 0.994i)7-s + (0.741 + 1.28i)11-s + (0.270 + 0.467i)13-s + (−0.502 + 0.871i)17-s + (0.0464 + 0.0803i)19-s + (0.287 − 0.498i)23-s + (0.488 + 0.845i)25-s + (−0.304 + 0.527i)29-s − 0.590·31-s + (−0.124 − 0.0901i)35-s + (−0.557 − 0.965i)37-s + (0.439 + 0.760i)41-s + (−0.756 + 1.31i)43-s + 1.92·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.760 - 0.649i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.760 - 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.786545092\)
\(L(\frac12)\) \(\approx\) \(1.786545092\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.271 + 2.63i)T \)
good5 \( 1 + (-0.171 + 0.297i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.45 - 4.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.974 - 1.68i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.07 - 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.202 - 0.350i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.37 + 2.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.63 - 2.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 + (3.38 + 5.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.81 - 4.87i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.96 - 8.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 + (1.38 - 2.39i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.45T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + 8.42T + 67T^{2} \)
71 \( 1 - 5.08T + 71T^{2} \)
73 \( 1 + (-4.58 + 7.93i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.49T + 79T^{2} \)
83 \( 1 + (4.62 - 8.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.54 + 6.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.31 - 2.27i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.173974153496262272915379951141, −8.400647599626878801474803377235, −7.32226159652310562367095629326, −6.97621093116188262255669733253, −6.10979748175727195119042072546, −4.95509281892594688748952962277, −4.23427495120076260789963789107, −3.58574363498050015768396848916, −2.08183657085976748906796313744, −1.19615059776040497359223367291, 0.69356706965403858913634434669, 2.16336160840658326891050112362, 3.07435463897296238537958186059, 3.93888057632287602864966744279, 5.18265073996543357696118258577, 5.73317539622918027831638036639, 6.53890948645803853405731401605, 7.33135047685179750528838335727, 8.560278740251796132056002038290, 8.688111576768916799139645091701

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