Properties

Label 2-1350-1.1-c3-0-24
Degree $2$
Conductor $1350$
Sign $1$
Analytic cond. $79.6525$
Root an. cond. $8.92482$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + 4·7-s + 8·8-s − 42·11-s − 20·13-s + 8·14-s + 16·16-s + 93·17-s + 59·19-s − 84·22-s + 9·23-s − 40·26-s + 16·28-s − 120·29-s + 47·31-s + 32·32-s + 186·34-s + 262·37-s + 118·38-s − 126·41-s + 178·43-s − 168·44-s + 18·46-s + 144·47-s − 327·49-s − 80·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.215·7-s + 0.353·8-s − 1.15·11-s − 0.426·13-s + 0.152·14-s + 1/4·16-s + 1.32·17-s + 0.712·19-s − 0.814·22-s + 0.0815·23-s − 0.301·26-s + 0.107·28-s − 0.768·29-s + 0.272·31-s + 0.176·32-s + 0.938·34-s + 1.16·37-s + 0.503·38-s − 0.479·41-s + 0.631·43-s − 0.575·44-s + 0.0576·46-s + 0.446·47-s − 0.953·49-s − 0.213·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(79.6525\)
Root analytic conductor: \(8.92482\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.435746157\)
\(L(\frac12)\) \(\approx\) \(3.435746157\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 + 42 T + p^{3} T^{2} \)
13 \( 1 + 20 T + p^{3} T^{2} \)
17 \( 1 - 93 T + p^{3} T^{2} \)
19 \( 1 - 59 T + p^{3} T^{2} \)
23 \( 1 - 9 T + p^{3} T^{2} \)
29 \( 1 + 120 T + p^{3} T^{2} \)
31 \( 1 - 47 T + p^{3} T^{2} \)
37 \( 1 - 262 T + p^{3} T^{2} \)
41 \( 1 + 126 T + p^{3} T^{2} \)
43 \( 1 - 178 T + p^{3} T^{2} \)
47 \( 1 - 144 T + p^{3} T^{2} \)
53 \( 1 - 741 T + p^{3} T^{2} \)
59 \( 1 - 444 T + p^{3} T^{2} \)
61 \( 1 - 221 T + p^{3} T^{2} \)
67 \( 1 - 538 T + p^{3} T^{2} \)
71 \( 1 + 690 T + p^{3} T^{2} \)
73 \( 1 - 1126 T + p^{3} T^{2} \)
79 \( 1 - 665 T + p^{3} T^{2} \)
83 \( 1 - 75 T + p^{3} T^{2} \)
89 \( 1 - 1086 T + p^{3} T^{2} \)
97 \( 1 + 1544 T + p^{3} T^{2} \)
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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

−9.403047305805344839122838368938, −8.087034733027938236516159384912, −7.66702847130679060421373638112, −6.76338228714600270939140875910, −5.51388039446001192140839301745, −5.28224078887667504905667824373, −4.10137778529032523346201001899, −3.10971591157685679855766932035, −2.23505131907932958020358425295, −0.836470127643280600879351906015, 0.836470127643280600879351906015, 2.23505131907932958020358425295, 3.10971591157685679855766932035, 4.10137778529032523346201001899, 5.28224078887667504905667824373, 5.51388039446001192140839301745, 6.76338228714600270939140875910, 7.66702847130679060421373638112, 8.087034733027938236516159384912, 9.403047305805344839122838368938

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