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Properties

Label 2-2268-63.25-c1-0-5
Degree $2$
Conductor $2268$
Sign $-0.488 - 0.872i$
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  + (0.705 − 1.22i)5-s + (−0.779 + 2.52i)7-s + (2.48 + 4.30i)11-s + (1.48 + 2.57i)13-s + (−1.29 + 2.25i)17-s + (−3.76 − 6.52i)19-s + (−4.06 + 7.03i)23-s + (1.50 + 2.60i)25-s + (3.46 − 6.00i)29-s − 10.5·31-s + (2.53 + 2.73i)35-s + (−0.0945 − 0.163i)37-s + (−1.02 − 1.76i)41-s + (2.19 − 3.79i)43-s − 9.38·47-s + ⋯
L(s)  = 1  + (0.315 − 0.546i)5-s + (−0.294 + 0.955i)7-s + (0.749 + 1.29i)11-s + (0.411 + 0.713i)13-s + (−0.315 + 0.546i)17-s + (−0.863 − 1.49i)19-s + (−0.847 + 1.46i)23-s + (0.301 + 0.521i)25-s + (0.644 − 1.11i)29-s − 1.89·31-s + (0.429 + 0.462i)35-s + (−0.0155 − 0.0269i)37-s + (−0.159 − 0.275i)41-s + (0.333 − 0.578i)43-s − 1.36·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.488 - 0.872i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.488 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.205911444\)
\(L(\frac12)\) \(\approx\) \(1.205911444\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.779 - 2.52i)T \)
good5 \( 1 + (-0.705 + 1.22i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.48 - 4.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.48 - 2.57i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.29 - 2.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.76 + 6.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.06 - 7.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.46 + 6.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + (0.0945 + 0.163i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.02 + 1.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.19 + 3.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.38T + 47T^{2} \)
53 \( 1 + (6.95 - 12.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.78T + 59T^{2} \)
61 \( 1 - 5.89T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + 7.86T + 71T^{2} \)
73 \( 1 + (-0.894 + 1.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.70T + 79T^{2} \)
83 \( 1 + (-0.0100 + 0.0174i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.33 - 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.33 - 7.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.254477503216846982105001721967, −8.816334592611317230647222742353, −7.75053246752463861015336124232, −6.81328407108746723706772200901, −6.22085789635948854010690967200, −5.30718078159830174705586273512, −4.50086402004293845477996495793, −3.67157241957804061921286432986, −2.24516427811096969173137832563, −1.63662360398118389407164648494, 0.39558309786492176854913937116, 1.72873802673884588296761697264, 3.13308234387541906910515806860, 3.66779338121098609451027578667, 4.65604152420262771015632849272, 5.92486065078335042055497028813, 6.37017413066600128672465411450, 7.05447155684789123043773358350, 8.205038798621355963126572348527, 8.541186085507350252450927292138

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