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Properties

Label 4-56e3-1.1-c1e2-0-16
Degree $4$
Conductor $175616$
Sign $1$
Analytic cond. $11.1974$
Root an. cond. $1.82927$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·5-s − 7-s − 2·9-s + 38·25-s − 8·31-s − 8·35-s + 16·43-s − 16·45-s + 8·47-s + 49-s − 8·61-s + 2·63-s − 24·67-s − 5·81-s − 24·101-s + 24·103-s − 24·107-s + 12·113-s − 22·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·155-s + ⋯
L(s)  = 1  + 3.57·5-s − 0.377·7-s − 2/3·9-s + 38/5·25-s − 1.43·31-s − 1.35·35-s + 2.43·43-s − 2.38·45-s + 1.16·47-s + 1/7·49-s − 1.02·61-s + 0.251·63-s − 2.93·67-s − 5/9·81-s − 2.38·101-s + 2.36·103-s − 2.32·107-s + 1.12·113-s − 2·121-s + 12.1·125-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 − 5.14·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(175616\)    =    \(2^{9} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(11.1974\)
Root analytic conductor: \(1.82927\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 175616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.041048216\)
\(L(\frac12)\) \(\approx\) \(3.041048216\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
7$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.5.ai_ba
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.61.i_fi
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.a_aby
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−9.271787274682453081680950548186, −8.870187268949863787955145092048, −8.676248321093563858087492291911, −7.51048678322729423104589438688, −7.22672629765934394974316502798, −6.28154185525062265042607550144, −6.27767574855681654030919817110, −5.61936194790385614498397188009, −5.57466416223045488855827312533, −4.90170998996837834018405216678, −4.05975063928059999821924523382, −2.87703403465824901690577931359, −2.66547510967810892791353271494, −1.94068717855646026245068222948, −1.30690906969982341922250798482, 1.30690906969982341922250798482, 1.94068717855646026245068222948, 2.66547510967810892791353271494, 2.87703403465824901690577931359, 4.05975063928059999821924523382, 4.90170998996837834018405216678, 5.57466416223045488855827312533, 5.61936194790385614498397188009, 6.27767574855681654030919817110, 6.28154185525062265042607550144, 7.22672629765934394974316502798, 7.51048678322729423104589438688, 8.676248321093563858087492291911, 8.870187268949863787955145092048, 9.271787274682453081680950548186

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