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Properties

Label 2-1458-9.4-c1-0-28
Degree $2$
Conductor $1458$
Sign $0.5 + 0.866i$
Analytic cond. $11.6421$
Root an. cond. $3.41206$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.400 − 0.692i)5-s + (−2.09 − 3.62i)7-s − 0.999·8-s + 0.800·10-s + (2.05 + 3.55i)11-s + (0.831 − 1.43i)13-s + (2.09 − 3.62i)14-s + (−0.5 − 0.866i)16-s − 2.09·17-s − 3.52·19-s + (0.400 + 0.692i)20-s + (−2.05 + 3.55i)22-s + (2.04 − 3.54i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.178 − 0.309i)5-s + (−0.791 − 1.37i)7-s − 0.353·8-s + 0.252·10-s + (0.619 + 1.07i)11-s + (0.230 − 0.399i)13-s + (0.559 − 0.968i)14-s + (−0.125 − 0.216i)16-s − 0.507·17-s − 0.808·19-s + (0.0894 + 0.154i)20-s + (−0.437 + 0.758i)22-s + (0.426 − 0.738i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1458\)    =    \(2 \cdot 3^{6}\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(11.6421\)
Root analytic conductor: \(3.41206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1458} (973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1458,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.391032892\)
\(L(\frac12)\) \(\approx\) \(1.391032892\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
good5 \( 1 + (-0.400 + 0.692i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.09 + 3.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.05 - 3.55i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.831 + 1.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.09T + 17T^{2} \)
19 \( 1 + 3.52T + 19T^{2} \)
23 \( 1 + (-2.04 + 3.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.34 + 7.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.18 + 7.25i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.49T + 37T^{2} \)
41 \( 1 + (-5.34 + 9.26i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.50 + 9.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.26 + 2.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.66T + 53T^{2} \)
59 \( 1 + (-5.79 + 10.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.60 + 2.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.47 - 2.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
79 \( 1 + (-2.00 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.897 - 1.55i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.80T + 89T^{2} \)
97 \( 1 + (-4.52 - 7.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.363001640824879541291172848862, −8.538813562884326748275983431528, −7.50566670526307155666677049325, −6.90499960049026296477917623512, −6.30990955652700934258516448632, −5.19619420781455840366379644036, −4.16397108783943062789627317583, −3.77208104752789565431293125296, −2.20467267705742652551032347167, −0.49497484039587805579854567613, 1.50611266968717468942901738496, 2.79404487692499621812670188321, 3.30088559757543159979074430636, 4.53151852043519684400333700812, 5.59854408395510128500142170219, 6.27100520397289719947112740844, 6.86032626666381175142888938488, 8.559111919159913026546396620341, 8.827679430431952981476176644304, 9.615570921513101517920959254215

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