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Properties

Label 2-2268-9.4-c1-0-14
Degree $2$
Conductor $2268$
Sign $0.173 + 0.984i$
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  + (−2 + 3.46i)5-s + (0.5 + 0.866i)7-s + (−1 − 1.73i)11-s + (3 − 5.19i)13-s − 4·17-s − 4·19-s + (−1 + 1.73i)23-s + (−5.49 − 9.52i)25-s + (1 + 1.73i)29-s − 3.99·35-s + 2·37-s + (2 + 3.46i)43-s + (−6 − 10.3i)47-s + (−0.499 + 0.866i)49-s − 6·53-s + ⋯
L(s)  = 1  + (−0.894 + 1.54i)5-s + (0.188 + 0.327i)7-s + (−0.301 − 0.522i)11-s + (0.832 − 1.44i)13-s − 0.970·17-s − 0.917·19-s + (−0.208 + 0.361i)23-s + (−1.09 − 1.90i)25-s + (0.185 + 0.321i)29-s − 0.676·35-s + 0.328·37-s + (0.304 + 0.528i)43-s + (−0.875 − 1.51i)47-s + (−0.0714 + 0.123i)49-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6483079835\)
\(L(\frac12)\) \(\approx\) \(0.6483079835\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.452445320215308997712389181311, −8.205117874874520281137929420529, −7.33160728326287922592009280461, −6.48540086436010212859951352585, −5.94992491857325362193599234821, −4.80409568455309079988321728165, −3.65201360089004495319850394693, −3.15863311698894899894797157765, −2.17742265112025065238118950789, −0.24271602992348347599577944285, 1.15292812845011734846901312917, 2.19041690873596747963118378836, 3.90156045537588755392256473343, 4.35650923732835223176343893425, 4.89126820607902665139450353310, 6.10929636377027391011311032325, 6.92145705418427138185999443698, 7.83623571249929562897794842347, 8.513209991261983443747381470336, 8.975304893681558232217073111616

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