Properties

Label 2-84150-1.1-c1-0-154
Degree $2$
Conductor $84150$
Sign $-1$
Analytic cond. $671.941$
Root an. cond. $25.9218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2·7-s + 8-s + 11-s − 4·13-s + 2·14-s + 16-s + 17-s + 2·19-s + 22-s + 2·23-s − 4·26-s + 2·28-s − 2·29-s − 4·31-s + 32-s + 34-s − 6·37-s + 2·38-s − 6·41-s − 2·43-s + 44-s + 2·46-s − 3·49-s − 4·52-s − 12·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 0.301·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.458·19-s + 0.213·22-s + 0.417·23-s − 0.784·26-s + 0.377·28-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.986·37-s + 0.324·38-s − 0.937·41-s − 0.304·43-s + 0.150·44-s + 0.294·46-s − 3/7·49-s − 0.554·52-s − 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(671.941\)
Root analytic conductor: \(25.9218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{84150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 84150,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.25994354514242, −13.87725875215171, −13.05331078494062, −12.79938910052893, −12.23630615049766, −11.67747011700412, −11.39583167085181, −10.88461069278080, −10.18587068717673, −9.794302857942565, −9.239429375693490, −8.523047155635506, −8.079430519992029, −7.474720811724098, −6.971406922776475, −6.634178464562333, −5.700507063601271, −5.306281586645802, −4.877626775734661, −4.349975906073586, −3.545853173951905, −3.196643988047403, −2.280744984784279, −1.834341706459346, −1.082060952635876, 0, 1.082060952635876, 1.834341706459346, 2.280744984784279, 3.196643988047403, 3.545853173951905, 4.349975906073586, 4.877626775734661, 5.306281586645802, 5.700507063601271, 6.634178464562333, 6.971406922776475, 7.474720811724098, 8.079430519992029, 8.523047155635506, 9.239429375693490, 9.794302857942565, 10.18587068717673, 10.88461069278080, 11.39583167085181, 11.67747011700412, 12.23630615049766, 12.79938910052893, 13.05331078494062, 13.87725875215171, 14.25994354514242

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