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Properties

Label 2-675-27.4-c1-0-51
Degree $2$
Conductor $675$
Sign $-0.835 + 0.549i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  + (0.826 + 0.300i)2-s + (−0.592 − 1.62i)3-s + (−0.939 − 0.788i)4-s − 1.52i·6-s + (2.87 − 2.41i)7-s + (−1.41 − 2.45i)8-s + (−2.29 + 1.92i)9-s + (−0.180 − 1.02i)11-s + (−0.726 + 1.99i)12-s + (−2.99 + 1.08i)13-s + (3.10 − 1.13i)14-s + (−0.00727 − 0.0412i)16-s + (0.233 − 0.405i)17-s + (−2.47 + 0.902i)18-s + (−2.34 − 4.06i)19-s + ⋯
L(s)  = 1  + (0.584 + 0.212i)2-s + (−0.342 − 0.939i)3-s + (−0.469 − 0.394i)4-s − 0.621i·6-s + (1.08 − 0.913i)7-s + (−0.501 − 0.868i)8-s + (−0.766 + 0.642i)9-s + (−0.0545 − 0.309i)11-s + (−0.209 + 0.576i)12-s + (−0.830 + 0.302i)13-s + (0.830 − 0.302i)14-s + (−0.00181 − 0.0103i)16-s + (0.0567 − 0.0982i)17-s + (−0.584 + 0.212i)18-s + (−0.538 − 0.932i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.835 + 0.549i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.354564 - 1.18432i\)
\(L(\frac12)\) \(\approx\) \(0.354564 - 1.18432i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.592 + 1.62i)T \)
5 \( 1 \)
good2 \( 1 + (-0.826 - 0.300i)T + (1.53 + 1.28i)T^{2} \)
7 \( 1 + (-2.87 + 2.41i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.180 + 1.02i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (2.99 - 1.08i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.233 + 0.405i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.34 + 4.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.11 - 3.45i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (5.45 + 1.98i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.14 + 2.63i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (2.23 - 3.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.52 - 2.73i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (2.11 + 11.9i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-2.65 + 2.22i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 8.83T + 53T^{2} \)
59 \( 1 + (-2.36 + 13.4i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-7.46 + 6.26i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.71 + 0.623i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (3.85 - 6.67i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.407 + 0.705i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.81 - 1.38i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-15.9 - 5.81i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-5.19 - 9.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.06 + 6.02i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.32571033903231957575504533795, −9.195713013298573971035362457778, −8.242100894859051867132825561355, −7.24235400752396300816685292314, −6.70542809336239920781890869585, −5.35485562563716158603113467226, −4.94934747095947924249752960568, −3.72529039646307801446107936313, −1.96743743901241239253370325181, −0.56867753597151268684574138849, 2.29294713796537206193624710606, 3.48430748220785210620397971537, 4.57355598820358008491044977073, 5.14516272183433376460249677986, 5.86965056779127845581559445539, 7.48025598289946419734313836695, 8.569564080621873707819527705566, 8.973754742740896181968551826176, 10.10013517463067973470823730292, 10.95043336631176467671598640072

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