Properties

Label 4-2816e2-1.1-c1e2-0-19
Degree $4$
Conductor $7929856$
Sign $1$
Analytic cond. $505.614$
Root an. cond. $4.74192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 3·9-s − 12·17-s + 2·23-s + 25-s + 14·31-s − 8·41-s + 16·47-s − 2·49-s + 12·63-s + 6·71-s − 32·73-s − 4·79-s − 30·89-s − 14·97-s − 32·103-s − 14·113-s + 48·119-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 36·153-s + 157-s + ⋯
L(s)  = 1  − 1.51·7-s − 9-s − 2.91·17-s + 0.417·23-s + 1/5·25-s + 2.51·31-s − 1.24·41-s + 2.33·47-s − 2/7·49-s + 1.51·63-s + 0.712·71-s − 3.74·73-s − 0.450·79-s − 3.17·89-s − 1.42·97-s − 3.15·103-s − 1.31·113-s + 4.40·119-s − 0.0909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.91·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7929856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7929856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7929856\)    =    \(2^{16} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(505.614\)
Root analytic conductor: \(4.74192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7929856,\ (\ :1/2, 1/2),\ 1)\)

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$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
11$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.951792919656583942452391720098, −8.232076787659667131500341637994, −8.153501340627254110874418683838, −7.25769700003516417665818433006, −6.91261142085879767085708962018, −6.72617148832644076232922190570, −6.43468283680466291160908087225, −5.99696449012798082260190639332, −5.60783085293647579536044915749, −5.18519311850200898399935449564, −4.50911498268406571824744437431, −4.22599600876556877852310622870, −4.01781540679499535849568740583, −2.96394313715142754796332552638, −2.90433129053123688060996573854, −2.68842288540089207453113789010, −1.92892948951150798705339609868, −1.13382715382671769874122575354, 0, 0, 1.13382715382671769874122575354, 1.92892948951150798705339609868, 2.68842288540089207453113789010, 2.90433129053123688060996573854, 2.96394313715142754796332552638, 4.01781540679499535849568740583, 4.22599600876556877852310622870, 4.50911498268406571824744437431, 5.18519311850200898399935449564, 5.60783085293647579536044915749, 5.99696449012798082260190639332, 6.43468283680466291160908087225, 6.72617148832644076232922190570, 6.91261142085879767085708962018, 7.25769700003516417665818433006, 8.153501340627254110874418683838, 8.232076787659667131500341637994, 8.951792919656583942452391720098

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