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Properties

Label 4-43904-1.1-c1e2-0-6
Degree $4$
Conductor $43904$
Sign $-1$
Analytic cond. $2.79935$
Root an. cond. $1.29349$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 7-s + 6·9-s + 4·19-s + 4·21-s + 6·25-s + 4·27-s + 4·29-s − 8·31-s − 12·37-s + 8·47-s + 49-s − 20·53-s − 16·57-s − 12·59-s − 6·63-s − 24·75-s − 37·81-s − 12·83-s − 16·87-s + 32·93-s + 24·103-s + 20·109-s + 48·111-s + 12·113-s − 22·121-s + 127-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.377·7-s + 2·9-s + 0.917·19-s + 0.872·21-s + 6/5·25-s + 0.769·27-s + 0.742·29-s − 1.43·31-s − 1.97·37-s + 1.16·47-s + 1/7·49-s − 2.74·53-s − 2.11·57-s − 1.56·59-s − 0.755·63-s − 2.77·75-s − 4.11·81-s − 1.31·83-s − 1.71·87-s + 3.31·93-s + 2.36·103-s + 1.91·109-s + 4.55·111-s + 1.12·113-s − 2·121-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(43904\)    =    \(2^{7} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(2.79935\)
Root analytic conductor: \(1.29349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 43904,\ (\ :1/2, 1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
7$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
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

−10.27322067133513101779746201704, −9.590662754018899629342276160434, −8.870187268949863787955145092048, −8.605229056688940905430426859414, −7.48523560915642098619060234168, −7.22672629765934394974316502798, −6.34948564084072494191115955004, −6.27767574855681654030919817110, −5.57466416223045488855827312533, −4.90170998996837834018405216678, −4.88658861407619853654072156509, −3.61118272831897977078492843408, −2.87703403465824901690577931359, −1.30690906969982341922250798482, 0, 1.30690906969982341922250798482, 2.87703403465824901690577931359, 3.61118272831897977078492843408, 4.88658861407619853654072156509, 4.90170998996837834018405216678, 5.57466416223045488855827312533, 6.27767574855681654030919817110, 6.34948564084072494191115955004, 7.22672629765934394974316502798, 7.48523560915642098619060234168, 8.605229056688940905430426859414, 8.870187268949863787955145092048, 9.590662754018899629342276160434, 10.27322067133513101779746201704

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