Properties

Label 2-35904-1.1-c1-0-66
Degree $2$
Conductor $35904$
Sign $-1$
Analytic cond. $286.694$
Root an. cond. $16.9320$
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 + 4·7-s + 9-s + 11-s − 6·13-s + 17-s − 6·19-s + 4·21-s − 5·25-s + 27-s + 2·29-s + 8·31-s + 33-s − 2·37-s − 6·39-s − 2·41-s − 2·43-s − 6·47-s + 9·49-s + 51-s + 2·53-s − 6·57-s − 2·59-s − 8·61-s + 4·63-s + 8·67-s − 4·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s + 0.242·17-s − 1.37·19-s + 0.872·21-s − 25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.174·33-s − 0.328·37-s − 0.960·39-s − 0.312·41-s − 0.304·43-s − 0.875·47-s + 9/7·49-s + 0.140·51-s + 0.274·53-s − 0.794·57-s − 0.260·59-s − 1.02·61-s + 0.503·63-s + 0.977·67-s − 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35904\)    =    \(2^{6} \cdot 3 \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(286.694\)
Root analytic conductor: \(16.9320\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 35904,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.03600855774345, −14.66407894356469, −14.24181284318533, −13.75914026597072, −13.19315438762938, −12.42456774639439, −11.99378546927266, −11.68101124168575, −10.83695728380466, −10.48228607640819, −9.703918851722199, −9.461952475851330, −8.476359970831475, −8.206550578370857, −7.827460269370602, −7.066654280358199, −6.611188427400323, −5.752663664476307, −5.011196689067806, −4.582163074858596, −4.138530202639184, −3.211503229637455, −2.358659711352755, −2.005374391872328, −1.199126177618042, 0, 1.199126177618042, 2.005374391872328, 2.358659711352755, 3.211503229637455, 4.138530202639184, 4.582163074858596, 5.011196689067806, 5.752663664476307, 6.611188427400323, 7.066654280358199, 7.827460269370602, 8.206550578370857, 8.476359970831475, 9.461952475851330, 9.703918851722199, 10.48228607640819, 10.83695728380466, 11.68101124168575, 11.99378546927266, 12.42456774639439, 13.19315438762938, 13.75914026597072, 14.24181284318533, 14.66407894356469, 15.03600855774345

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