Properties

Label 2-2268-63.25-c1-0-2
Degree $2$
Conductor $2268$
Sign $-0.999 + 0.0137i$
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  + (−1.21 + 2.09i)5-s + (−2.56 + 0.658i)7-s + (2.35 + 4.07i)11-s + (1.71 + 2.96i)13-s + (0.851 − 1.47i)17-s + (−0.641 − 1.11i)19-s + (−0.562 + 0.974i)23-s + (−0.430 − 0.746i)25-s + (−2.35 + 4.07i)29-s − 3.42·31-s + (1.72 − 6.17i)35-s + (−4.27 − 7.40i)37-s + (1.85 + 3.21i)41-s + (−2.77 + 4.80i)43-s + 11.8·47-s + ⋯
L(s)  = 1  + (−0.541 + 0.937i)5-s + (−0.968 + 0.249i)7-s + (0.709 + 1.22i)11-s + (0.474 + 0.821i)13-s + (0.206 − 0.357i)17-s + (−0.147 − 0.254i)19-s + (−0.117 + 0.203i)23-s + (−0.0861 − 0.149i)25-s + (−0.436 + 0.756i)29-s − 0.614·31-s + (0.290 − 1.04i)35-s + (−0.702 − 1.21i)37-s + (0.290 + 0.502i)41-s + (−0.422 + 0.732i)43-s + 1.72·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.999 + 0.0137i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.999 + 0.0137i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7290078871\)
\(L(\frac12)\) \(\approx\) \(0.7290078871\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.56 - 0.658i)T \)
good5 \( 1 + (1.21 - 2.09i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.35 - 4.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.71 - 2.96i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.851 + 1.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.562 - 0.974i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.35 - 4.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.42T + 31T^{2} \)
37 \( 1 + (4.27 + 7.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.85 - 3.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.77 - 4.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + (-5.13 + 8.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.12T + 59T^{2} \)
61 \( 1 + 9.24T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + (1.06 - 1.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 + (-4.21 + 7.29i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.04 - 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.12 - 14.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.212349590987730715686763480699, −8.989866887587230540141348728181, −7.51959014414942919356857355725, −7.10131549824887963654916750425, −6.51694999358424265956520437937, −5.61318963431816277589101068084, −4.37478347848738606756499919504, −3.68767062410425651453073321205, −2.83842132311884257877032828803, −1.70141834636067025403141872588, 0.26950121376075235611103546419, 1.26076312238113980114004971604, 2.97857613145833086994926594131, 3.73091622869532737835609344346, 4.41018225758973032244684998825, 5.76836904871349476009330967404, 6.00940236084481602884015887771, 7.16278862598787907142472288147, 7.973225082313703173868577023513, 8.817413542073001686054376134139

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