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Properties

Label 2-1350-9.7-c1-0-12
Degree $2$
Conductor $1350$
Sign $0.766 + 0.642i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (3 − 5.19i)11-s + (1 + 1.73i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 4·19-s + (3 + 5.19i)22-s + (−4.5 − 7.79i)23-s − 1.99·26-s + 0.999·28-s + (1.5 − 2.59i)29-s + (2 + 3.46i)31-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (0.904 − 1.56i)11-s + (0.277 + 0.480i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s − 0.917·19-s + (0.639 + 1.10i)22-s + (−0.938 − 1.62i)23-s − 0.392·26-s + 0.188·28-s + (0.278 − 0.482i)29-s + (0.359 + 0.622i)31-s + (−0.0883 − 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.098505726\)
\(L(\frac12)\) \(\approx\) \(1.098505726\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.164333772642387192789771684480, −8.673643113194546238716509087692, −8.172200978028912142730991282443, −6.83451291510766348085593582433, −6.33679087051504731293784758384, −5.64246077557228315658256775443, −4.41406478238854838983456340249, −3.53138756085233437597713095755, −2.11515490376249954656784950438, −0.54995543056266797734947930049, 1.31673424141656041364256313145, 2.32873392221335366537362804982, 3.70091175476191681069026203336, 4.27563843815304877202730897040, 5.44213448760577915414703800134, 6.61390421211242479918307628743, 7.31090421310699097285698098629, 8.171852383028030900330102108034, 9.065483656997210116745164352971, 9.859174724820605227048372779751

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