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Properties

Label 2-1350-45.38-c1-0-9
Degree $2$
Conductor $1350$
Sign $0.0712 + 0.997i$
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 + (0.622 + 2.32i)7-s + (−0.707 + 0.707i)8-s + (−0.991 − 0.572i)11-s + (0.640 − 2.38i)13-s + (−1.20 − 2.08i)14-s + (0.500 − 0.866i)16-s + (−4.99 − 4.99i)17-s − 2.78i·19-s + (1.10 + 0.296i)22-s + (−5.95 − 1.59i)23-s + 2.47i·26-s + (1.70 + 1.70i)28-s + (−0.672 + 1.16i)29-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (0.235 + 0.877i)7-s + (−0.249 + 0.249i)8-s + (−0.299 − 0.172i)11-s + (0.177 − 0.662i)13-s + (−0.321 − 0.556i)14-s + (0.125 − 0.216i)16-s + (−1.21 − 1.21i)17-s − 0.638i·19-s + (0.235 + 0.0631i)22-s + (−1.24 − 0.332i)23-s + 0.485i·26-s + (0.321 + 0.321i)28-s + (−0.124 + 0.216i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0712 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0712 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.0712 + 0.997i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.0712 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7323820118\)
\(L(\frac12)\) \(\approx\) \(0.7323820118\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.622 - 2.32i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.991 + 0.572i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.640 + 2.38i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (4.99 + 4.99i)T + 17iT^{2} \)
19 \( 1 + 2.78iT - 19T^{2} \)
23 \( 1 + (5.95 + 1.59i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.672 - 1.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.16 + 8.16i)T - 37iT^{2} \)
41 \( 1 + (-1.70 + 0.986i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.68 - 2.32i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-11.9 + 3.19i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.84 + 1.84i)T - 53iT^{2} \)
59 \( 1 + (1.31 + 2.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.54 - 6.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0545 + 0.0146i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 9.10iT - 71T^{2} \)
73 \( 1 + (7.82 + 7.82i)T + 73iT^{2} \)
79 \( 1 + (8.46 + 4.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.724 - 2.70i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 4.87T + 89T^{2} \)
97 \( 1 + (2.08 + 7.79i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.164855975499504009793484619199, −8.763386317003401555853772095470, −7.88081735177502308623965201234, −7.10950044521915376072986417283, −6.12429274214732878968568152833, −5.40773942565298056675242066830, −4.40721461465608848145581524701, −2.88265354428871388775166562106, −2.12402431224971175903293980656, −0.38024492805805096473821498343, 1.35820431481939865066815580498, 2.37784213589025971068750644965, 3.87852532246914178380269845520, 4.40401356536364897968027248467, 5.91517459782280821786421751824, 6.60756725766799921030681941835, 7.57620322961364747189783330793, 8.172797399505620680059739078841, 8.965404486224502670184676135791, 9.963655329375682359210224941851

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