Properties

Label 4-1859e2-1.1-c1e2-0-1
Degree $4$
Conductor $3455881$
Sign $1$
Analytic cond. $220.349$
Root an. cond. $3.85281$
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  − 2·3-s − 4-s + 6·7-s + 2·11-s + 2·12-s − 3·16-s − 12·17-s − 6·19-s − 12·21-s − 7·25-s + 2·27-s − 6·28-s + 12·29-s − 12·31-s − 4·33-s − 18·41-s − 4·43-s − 2·44-s + 6·47-s + 6·48-s + 16·49-s + 24·51-s + 6·53-s + 12·57-s − 6·59-s − 4·61-s + 7·64-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 2.26·7-s + 0.603·11-s + 0.577·12-s − 3/4·16-s − 2.91·17-s − 1.37·19-s − 2.61·21-s − 7/5·25-s + 0.384·27-s − 1.13·28-s + 2.22·29-s − 2.15·31-s − 0.696·33-s − 2.81·41-s − 0.609·43-s − 0.301·44-s + 0.875·47-s + 0.866·48-s + 16/7·49-s + 3.36·51-s + 0.824·53-s + 1.58·57-s − 0.781·59-s − 0.512·61-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3455881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3455881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3455881\)    =    \(11^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(220.349\)
Root analytic conductor: \(3.85281\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3455881,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad11$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.7.ag_u
17$D_{4}$ \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.17.m_cp
19$D_{4}$ \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bs
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$D_{4}$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_dn
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
37$C_2^2$ \( 1 + 71 T^{2} + p^{2} T^{4} \) 2.37.a_ct
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.41.s_gh
43$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_as
47$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_dw
53$D_{4}$ \( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_dz
59$D_{4}$ \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_eu
61$D_{4}$ \( 1 + 4 T + 99 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_dv
67$D_{4}$ \( 1 + 6 T + 140 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_fk
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.71.a_fa
73$D_{4}$ \( 1 + 18 T + 215 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.73.s_ih
79$D_{4}$ \( 1 - 2 T + 132 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_fc
83$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_bc
89$C_2^2$ \( 1 + 166 T^{2} + p^{2} T^{4} \) 2.89.a_gk
97$C_2^2$ \( 1 + 182 T^{2} + p^{2} T^{4} \) 2.97.a_ha
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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

−8.804775764886987705445094143874, −8.665372750416213907539561418447, −8.259336258444610416432827588953, −8.233816182281546400996346950363, −7.28209130680040409253173349116, −7.04071382361689778524739705397, −6.54934328305350283824644262972, −6.37193216055907962674294408693, −5.64233885191080285053443132873, −5.42150133634244089322636797548, −4.88831870263982615052864886581, −4.48878707395601464055763607997, −4.30703374536162876605522674923, −4.04273610866298544541203467196, −3.02688104177703958292730813871, −2.16287402293091349196595794496, −1.92861885875819579693784452552, −1.44958012664702838224256496010, 0, 0, 1.44958012664702838224256496010, 1.92861885875819579693784452552, 2.16287402293091349196595794496, 3.02688104177703958292730813871, 4.04273610866298544541203467196, 4.30703374536162876605522674923, 4.48878707395601464055763607997, 4.88831870263982615052864886581, 5.42150133634244089322636797548, 5.64233885191080285053443132873, 6.37193216055907962674294408693, 6.54934328305350283824644262972, 7.04071382361689778524739705397, 7.28209130680040409253173349116, 8.233816182281546400996346950363, 8.259336258444610416432827588953, 8.665372750416213907539561418447, 8.804775764886987705445094143874

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