Properties

Label 2-162624-1.1-c1-0-140
Degree $2$
Conductor $162624$
Sign $-1$
Analytic cond. $1298.55$
Root an. cond. $36.0355$
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  + 3-s − 3·5-s − 7-s + 9-s − 4·13-s − 3·15-s + 7·17-s − 6·19-s − 21-s + 4·23-s + 4·25-s + 27-s + 2·29-s + 8·31-s + 3·35-s + 6·37-s − 4·39-s − 6·41-s − 9·43-s − 3·45-s − 3·47-s + 49-s + 7·51-s + 6·53-s − 6·57-s + 3·59-s + 6·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.774·15-s + 1.69·17-s − 1.37·19-s − 0.218·21-s + 0.834·23-s + 4/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.507·35-s + 0.986·37-s − 0.640·39-s − 0.937·41-s − 1.37·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s + 0.980·51-s + 0.824·53-s − 0.794·57-s + 0.390·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162624\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1298.55\)
Root analytic conductor: \(36.0355\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{162624} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 162624,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−13.31704521040547, −13.05507307727580, −12.49996428608119, −12.07089746889039, −11.66819184406189, −11.36442707716960, −10.40406600591639, −10.10908968376546, −9.907709764477782, −9.040127014449940, −8.571709295818806, −8.211492474500365, −7.728027931422590, −7.263141183963036, −6.875170509422681, −6.246669982704268, −5.606511554618212, −4.777918058342232, −4.579914398714198, −3.926320629244283, −3.310749071533678, −2.964269685363817, −2.388489059639316, −1.487705501448943, −0.7359844806849728, 0, 0.7359844806849728, 1.487705501448943, 2.388489059639316, 2.964269685363817, 3.310749071533678, 3.926320629244283, 4.579914398714198, 4.777918058342232, 5.606511554618212, 6.246669982704268, 6.875170509422681, 7.263141183963036, 7.728027931422590, 8.211492474500365, 8.571709295818806, 9.040127014449940, 9.907709764477782, 10.10908968376546, 10.40406600591639, 11.36442707716960, 11.66819184406189, 12.07089746889039, 12.49996428608119, 13.05507307727580, 13.31704521040547

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