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Properties

Label 2-1458-9.4-c1-0-10
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 + (1.04 − 1.80i)5-s + (2.00 + 3.47i)7-s − 0.999·8-s + 2.08·10-s + (0.106 + 0.183i)11-s + (−2.59 + 4.50i)13-s + (−2.00 + 3.47i)14-s + (−0.5 − 0.866i)16-s − 3.78·17-s − 1.27·19-s + (1.04 + 1.80i)20-s + (−0.106 + 0.183i)22-s + (0.414 − 0.718i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.466 − 0.808i)5-s + (0.758 + 1.31i)7-s − 0.353·8-s + 0.660·10-s + (0.0320 + 0.0554i)11-s + (−0.720 + 1.24i)13-s + (−0.536 + 0.928i)14-s + (−0.125 − 0.216i)16-s − 0.918·17-s − 0.292·19-s + (0.233 + 0.404i)20-s + (−0.0226 + 0.0392i)22-s + (0.0864 − 0.149i)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

\(L(1)\) \(\approx\) \(1.949025955\)
\(L(\frac12)\) \(\approx\) \(1.949025955\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 + 1.80i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.00 - 3.47i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.106 - 0.183i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.59 - 4.50i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.78T + 17T^{2} \)
19 \( 1 + 1.27T + 19T^{2} \)
23 \( 1 + (-0.414 + 0.718i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.60 - 7.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.76 - 3.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.55T + 37T^{2} \)
41 \( 1 + (-1.36 + 2.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.44 + 5.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.638 + 1.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (4.27 - 7.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.41 + 2.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.88 - 8.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 - 0.750T + 73T^{2} \)
79 \( 1 + (-1.18 - 2.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.70 + 9.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.39T + 89T^{2} \)
97 \( 1 + (-6.28 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.299251193928295421680739820224, −8.890553505948518062567079705191, −8.437055731666901838241275837065, −7.20744202131479136450651076465, −6.50712287161537171527434160296, −5.42997058054116123864540563684, −4.99893066166067035895293897725, −4.20285876697378291578319449228, −2.59971138337217501730310149174, −1.69256584168547360141006746311, 0.68320757229611874682170061096, 2.14293535201261628983939946411, 2.99689929154920178756280642365, 4.16552176275685986008884193002, 4.80491569286512209499608820133, 5.93606263760306462767614764637, 6.75610957921393164193475906855, 7.65243902499529374769035361642, 8.328787135131514433781030764992, 9.711120976295053330518373401734

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