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Properties

Label 4-2624e2-1.1-c1e2-0-15
Degree $4$
Conductor $6885376$
Sign $1$
Analytic cond. $439.017$
Root an. cond. $4.57741$
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 − 4·9-s + 4·17-s + 8·19-s + 8·23-s − 2·25-s − 8·29-s − 8·31-s − 2·41-s − 8·43-s − 4·47-s − 24·53-s + 8·59-s − 12·61-s + 16·63-s + 8·67-s − 4·71-s − 16·73-s − 12·79-s + 7·81-s − 24·83-s − 12·89-s − 4·97-s + 24·101-s − 8·103-s − 24·107-s − 24·109-s + ⋯
L(s)  = 1  − 1.51·7-s − 4/3·9-s + 0.970·17-s + 1.83·19-s + 1.66·23-s − 2/5·25-s − 1.48·29-s − 1.43·31-s − 0.312·41-s − 1.21·43-s − 0.583·47-s − 3.29·53-s + 1.04·59-s − 1.53·61-s + 2.01·63-s + 0.977·67-s − 0.474·71-s − 1.87·73-s − 1.35·79-s + 7/9·81-s − 2.63·83-s − 1.27·89-s − 0.406·97-s + 2.38·101-s − 0.788·103-s − 2.32·107-s − 2.29·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6885376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6885376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6885376\)    =    \(2^{12} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(439.017\)
Root analytic conductor: \(4.57741\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6885376,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
41$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_q
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_g
19$D_{4}$ \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.19.ai_ca
23$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_cc
29$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_bq
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cs
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.37.a_c
43$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cs
47$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_bw
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_ew
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$D_{4}$ \( 1 - 8 T + 132 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_fc
71$D_{4}$ \( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_fo
73$D_{4}$ \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_gw
79$D_{4}$ \( 1 + 12 T + 176 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_gu
83$D_{4}$ \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.83.y_ks
89$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_ha
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_gk
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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

−8.556663041382392756824714247761, −8.541953787527448209707744874318, −7.73470141730220440928295714203, −7.59905730042992697795225257523, −7.17955866304031337127281458626, −6.77397563821530031645039736757, −6.36181881034977596301464660938, −5.91484347307399301489398802493, −5.56238622689542331379141783845, −5.26578949564423614388484675900, −4.98569948500582184650356251287, −4.18983320898763646485839476448, −3.61035793175281693608996000809, −3.23500739525930494639628431598, −2.91597186512198263584908459667, −2.89162544690013129044783400702, −1.59240422266468154245190160125, −1.39879954868112891871178418409, 0, 0, 1.39879954868112891871178418409, 1.59240422266468154245190160125, 2.89162544690013129044783400702, 2.91597186512198263584908459667, 3.23500739525930494639628431598, 3.61035793175281693608996000809, 4.18983320898763646485839476448, 4.98569948500582184650356251287, 5.26578949564423614388484675900, 5.56238622689542331379141783845, 5.91484347307399301489398802493, 6.36181881034977596301464660938, 6.77397563821530031645039736757, 7.17955866304031337127281458626, 7.59905730042992697795225257523, 7.73470141730220440928295714203, 8.541953787527448209707744874318, 8.556663041382392756824714247761

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