Properties

Label 2-39984-1.1-c1-0-88
Degree $2$
Conductor $39984$
Sign $-1$
Analytic cond. $319.273$
Root an. cond. $17.8682$
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 + 2·5-s + 9-s + 6·13-s − 2·15-s − 17-s + 8·23-s − 25-s − 27-s − 6·29-s − 8·31-s + 10·37-s − 6·39-s + 6·41-s − 12·43-s + 2·45-s + 51-s − 10·53-s − 8·59-s − 6·61-s + 12·65-s − 12·67-s − 8·69-s + 6·73-s + 75-s + 8·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.66·13-s − 0.516·15-s − 0.242·17-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.960·39-s + 0.937·41-s − 1.82·43-s + 0.298·45-s + 0.140·51-s − 1.37·53-s − 1.04·59-s − 0.768·61-s + 1.48·65-s − 1.46·67-s − 0.963·69-s + 0.702·73-s + 0.115·75-s + 0.900·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39984\)    =    \(2^{4} \cdot 3 \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(319.273\)
Root analytic conductor: \(17.8682\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{39984} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 39984,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.05321446187649, −14.60930529477696, −13.81078965575082, −13.41992216027841, −12.97065873710832, −12.72076592488550, −11.75245090680044, −11.27700149848067, −10.84346580705203, −10.56606359827229, −9.656736406251172, −9.146506335105158, −9.013524877742196, −7.972356152627851, −7.582701399159548, −6.738631377994659, −6.217039481478766, −5.934280437774619, −5.212748399956355, −4.733054392187898, −3.828771656301505, −3.354992931069669, −2.457674062743786, −1.580237168073311, −1.189180276005915, 0, 1.189180276005915, 1.580237168073311, 2.457674062743786, 3.354992931069669, 3.828771656301505, 4.733054392187898, 5.212748399956355, 5.934280437774619, 6.217039481478766, 6.738631377994659, 7.582701399159548, 7.972356152627851, 9.013524877742196, 9.146506335105158, 9.656736406251172, 10.56606359827229, 10.84346580705203, 11.27700149848067, 11.75245090680044, 12.72076592488550, 12.97065873710832, 13.41992216027841, 13.81078965575082, 14.60930529477696, 15.05321446187649

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