Properties

Degree $4$
Conductor $14745600$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s − 9-s + 12·17-s − 25-s − 16·31-s + 12·41-s + 34·49-s + 8·63-s − 4·73-s − 16·79-s + 81-s − 36·89-s + 4·97-s − 8·103-s − 36·113-s − 96·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3.02·7-s − 1/3·9-s + 2.91·17-s − 1/5·25-s − 2.87·31-s + 1.87·41-s + 34/7·49-s + 1.00·63-s − 0.468·73-s − 1.80·79-s + 1/9·81-s − 3.81·89-s + 0.406·97-s − 0.788·103-s − 3.38·113-s − 8.80·119-s + 2·121-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 − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3354680708\)
\(L(\frac12)\) \(\approx\) \(0.3354680708\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−9.049501633708895475355200817898, −8.074728951994574644427749423187, −8.041281451502253573779367608135, −7.36781763279993281526828451162, −7.27986764413298653869240839105, −6.81715580327141751355009472148, −6.51345380999056069954935343856, −5.89931564055475593586577752845, −5.69204567733969071922162444918, −5.62046007723335698577820786912, −5.18374109706472440891985596441, −4.13367776901290376629116522757, −4.03695833279852709258724400268, −3.44081796995605314524539879977, −3.33838303718534531046498251516, −2.70367902591892400586404656081, −2.69592201268720051595776666302, −1.60093197994726877772893579022, −1.06792132574825362115703732758, −0.18873018619139776894527362869, 0.18873018619139776894527362869, 1.06792132574825362115703732758, 1.60093197994726877772893579022, 2.69592201268720051595776666302, 2.70367902591892400586404656081, 3.33838303718534531046498251516, 3.44081796995605314524539879977, 4.03695833279852709258724400268, 4.13367776901290376629116522757, 5.18374109706472440891985596441, 5.62046007723335698577820786912, 5.69204567733969071922162444918, 5.89931564055475593586577752845, 6.51345380999056069954935343856, 6.81715580327141751355009472148, 7.27986764413298653869240839105, 7.36781763279993281526828451162, 8.041281451502253573779367608135, 8.074728951994574644427749423187, 9.049501633708895475355200817898

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