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Properties

Label 2-1350-45.4-c1-0-4
Degree $2$
Conductor $1350$
Sign $0.893 - 0.448i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.05 + 1.18i)7-s − 0.999i·8-s + (0.686 + 1.18i)11-s + (4.10 + 2.37i)13-s + (−1.18 + 2.05i)14-s + (−0.5 − 0.866i)16-s + 7.37i·17-s − 3.37·19-s + (1.18 + 0.686i)22-s + (3.78 + 2.18i)23-s + 4.74·26-s + 2.37i·28-s + (2.18 + 3.78i)29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.776 + 0.448i)7-s − 0.353i·8-s + (0.206 + 0.358i)11-s + (1.13 + 0.657i)13-s + (−0.317 + 0.549i)14-s + (−0.125 − 0.216i)16-s + 1.78i·17-s − 0.773·19-s + (0.253 + 0.146i)22-s + (0.789 + 0.455i)23-s + 0.930·26-s + 0.448i·28-s + (0.405 + 0.703i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.893 - 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.167800765\)
\(L(\frac12)\) \(\approx\) \(2.167800765\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (2.05 - 1.18i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.686 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.10 - 2.37i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.37iT - 17T^{2} \)
19 \( 1 + 3.37T + 19T^{2} \)
23 \( 1 + (-3.78 - 2.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.18 - 3.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.37 + 5.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.84 + 5.68i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.40 + 0.813i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + (-0.686 + 1.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.55 + 7.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.06 - 3.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 14.1iT - 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.40 + 0.813i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.11T + 89T^{2} \)
97 \( 1 + (2.27 - 1.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.655564331335217514863788868070, −8.982828843019698987331232043361, −8.178601038798406201623270177181, −6.89861347182866643397676746632, −6.23862069079492499423772586305, −5.63607162457270068046697996361, −4.27047312093410542094481284674, −3.74216918410172504472015567338, −2.56204006399005974159568528600, −1.42456103029138621281479057331, 0.78761058339001017855222096227, 2.72389218972973398508018893384, 3.43393777991605547843324304313, 4.45766706599729517582395146045, 5.36475147665900519603041146317, 6.40387586715244915358248047649, 6.79395900798750180674997856620, 7.85047323132925501619313444751, 8.659833865146007964321356301386, 9.474531574113664049643315477303

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