Properties

Label 2-2268-63.25-c1-0-16
Degree $2$
Conductor $2268$
Sign $0.792 + 0.610i$
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  + (−0.515 + 0.892i)5-s + (−2.63 + 0.277i)7-s + (0.792 + 1.37i)11-s + (−2.52 − 4.37i)13-s + (−2.58 + 4.47i)17-s + (−0.392 − 0.680i)19-s + (2.93 − 5.07i)23-s + (1.96 + 3.40i)25-s + (4.44 − 7.69i)29-s − 1.15·31-s + (1.10 − 2.49i)35-s + (4.07 + 7.06i)37-s + (−3.87 − 6.70i)41-s + (1.26 − 2.19i)43-s + 8.49·47-s + ⋯
L(s)  = 1  + (−0.230 + 0.399i)5-s + (−0.994 + 0.104i)7-s + (0.239 + 0.414i)11-s + (−0.700 − 1.21i)13-s + (−0.626 + 1.08i)17-s + (−0.0901 − 0.156i)19-s + (0.611 − 1.05i)23-s + (0.393 + 0.681i)25-s + (0.825 − 1.42i)29-s − 0.206·31-s + (0.187 − 0.421i)35-s + (0.670 + 1.16i)37-s + (−0.604 − 1.04i)41-s + (0.193 − 0.334i)43-s + 1.23·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.792 + 0.610i$
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: \((2,\ 2268,\ (\ :1/2),\ 0.792 + 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.174655553\)
\(L(\frac12)\) \(\approx\) \(1.174655553\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 0.277i)T \)
good5 \( 1 + (0.515 - 0.892i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.792 - 1.37i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.52 + 4.37i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.58 - 4.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.392 + 0.680i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.93 + 5.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.44 + 7.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.15T + 31T^{2} \)
37 \( 1 + (-4.07 - 7.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.87 + 6.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.26 + 2.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.49T + 47T^{2} \)
53 \( 1 + (-2.41 + 4.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.87T + 59T^{2} \)
61 \( 1 + 9.64T + 61T^{2} \)
67 \( 1 + 1.67T + 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + (-3.04 + 5.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 8.31T + 79T^{2} \)
83 \( 1 + (-7.12 + 12.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.69 - 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.67 - 4.63i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.928766265224725342090256563036, −8.191351226164523067661888494762, −7.30181633801727951209093701588, −6.62351154889993574396226990800, −5.96337620463161530852589654091, −4.92620900117079722682179645162, −3.99149335023996950429400039996, −3.06118579724547976904613150538, −2.29418575830759712962846517191, −0.52392702682741311351917898977, 0.906197546659440586437137710049, 2.39682957670781539918555389169, 3.31897351277874503279021946303, 4.30991802748541391686751186817, 5.01678830670439544165098089992, 6.06933494408507438787007140511, 6.90921366026835989246900323682, 7.31827346975281506681901071967, 8.523806358262134797418045892975, 9.247332851656990097571739430823

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