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Properties

Label 2-1350-45.38-c1-0-11
Degree $2$
Conductor $1350$
Sign $0.724 + 0.688i$
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 + (−0.283 − 1.05i)7-s + (0.707 − 0.707i)8-s + (5.44 + 3.14i)11-s + (0.896 − 3.34i)13-s + (−0.548 − 0.949i)14-s + (0.500 − 0.866i)16-s + (−3.14 − 3.14i)17-s − 1.55i·19-s + (6.07 + 1.62i)22-s + (0.965 + 0.258i)23-s − 3.46i·26-s + (−0.775 − 0.775i)28-s + (−1.57 + 2.72i)29-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (−0.107 − 0.400i)7-s + (0.249 − 0.249i)8-s + (1.64 + 0.948i)11-s + (0.248 − 0.928i)13-s + (−0.146 − 0.253i)14-s + (0.125 − 0.216i)16-s + (−0.763 − 0.763i)17-s − 0.355i·19-s + (1.29 + 0.347i)22-s + (0.201 + 0.0539i)23-s − 0.679i·26-s + (−0.146 − 0.146i)28-s + (−0.292 + 0.505i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.724 + 0.688i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.724 + 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.740504217\)
\(L(\frac12)\) \(\approx\) \(2.740504217\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.283 + 1.05i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-5.44 - 3.14i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.896 + 3.34i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (3.14 + 3.14i)T + 17iT^{2} \)
19 \( 1 + 1.55iT - 19T^{2} \)
23 \( 1 + (-0.965 - 0.258i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.57 - 2.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.22 - 3.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + (-3.39 + 1.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.34 + 0.896i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-8.69 + 2.32i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.61 - 6.61i)T - 53iT^{2} \)
59 \( 1 + (5.90 + 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.72 + 4.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.65 - 0.978i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.635iT - 71T^{2} \)
73 \( 1 + (2.89 + 2.89i)T + 73iT^{2} \)
79 \( 1 + (2.12 + 1.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.142 + 0.531i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.36T + 89T^{2} \)
97 \( 1 + (-2.89 - 10.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.453609150546751573547989115936, −8.949013655333708534875939152121, −7.61624226817383665610767987665, −6.93803599506718954057338767598, −6.26004695373000813825139074863, −5.14980892135988823151320631659, −4.32153597637289061186491179645, −3.54807816935979621337690226241, −2.37064444846396456101278176397, −1.05182029421878060771845516098, 1.41554405197189263692691217590, 2.68234352225653418632789505284, 3.94907392253559659640883509944, 4.30698096946346691793609819085, 5.82854074873320824386619845740, 6.21337690456789406333812874087, 6.98106571874888686376858993164, 8.132101255634715785855276529649, 8.919398259268915375963565443573, 9.474019460174039969538431526314

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