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Properties

Label 2-1350-45.38-c1-0-3
Degree $2$
Conductor $1350$
Sign $0.929 - 0.370i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.896 − 3.34i)7-s + (−0.707 + 0.707i)8-s + (1.5 + 0.866i)11-s + (−0.896 + 3.34i)13-s + (1.73 + 3.00i)14-s + (0.500 − 0.866i)16-s + (2.12 + 2.12i)17-s + 7i·19-s + (−1.67 − 0.448i)22-s + (−5.79 − 1.55i)23-s − 3.46i·26-s + (−2.44 − 2.44i)28-s + (1.73 − 3i)29-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (−0.338 − 1.26i)7-s + (−0.249 + 0.249i)8-s + (0.452 + 0.261i)11-s + (−0.248 + 0.928i)13-s + (0.462 + 0.801i)14-s + (0.125 − 0.216i)16-s + (0.514 + 0.514i)17-s + 1.60i·19-s + (−0.356 − 0.0955i)22-s + (−1.20 − 0.323i)23-s − 0.679i·26-s + (−0.462 − 0.462i)28-s + (0.321 − 0.557i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.929 - 0.370i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.929 - 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.122110660\)
\(L(\frac12)\) \(\approx\) \(1.122110660\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.896 + 3.34i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.896 - 3.34i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.12 - 2.12i)T + 17iT^{2} \)
19 \( 1 - 7iT - 19T^{2} \)
23 \( 1 + (5.79 + 1.55i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.73 + 3i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \)
41 \( 1 + (-10.5 + 6.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.01 + 1.34i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-5.79 + 1.55i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + (-6.06 - 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.67 + 0.448i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (6.12 + 6.12i)T + 73iT^{2} \)
79 \( 1 + (-3.46 - 2i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.10 - 11.5i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 + (-2.24 - 8.36i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.745684819443533059659574662743, −8.903939050578796905832308981055, −7.894707861610963936579130337199, −7.38401673219290050915923249110, −6.47953110456562501761133851420, −5.81727874194854769028663534770, −4.28492593988553052706899004826, −3.77514477809048064602576932030, −2.16614880284205199443177110876, −0.976686138621032752247737053073, 0.76489559445784574508932143395, 2.46016761592660748385908540592, 2.98435073269791264841752959782, 4.42236885602269871215604413550, 5.63535142533146765009162435363, 6.19237656630038376466746020635, 7.28911591709780018773384041337, 8.091549889924627875759955050758, 8.844920583647166381470565913767, 9.559216188901794885891029374666

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