↑

Properties

Label 4-340e2-1.1-c1e2-0-19
Degree $4$
Conductor $115600$
Sign $-1$
Analytic cond. $7.37075$
Root an. cond. $1.64769$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 5-s − 4·9-s − 2·10-s − 4·16-s − 3·17-s − 8·18-s − 2·20-s − 4·25-s − 9·29-s − 8·32-s − 6·34-s − 8·36-s + 5·37-s − 3·41-s + 4·45-s + 4·49-s − 8·50-s − 10·53-s − 18·58-s − 2·61-s − 8·64-s − 6·68-s − 7·73-s + 10·74-s + 4·80-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.447·5-s − 4/3·9-s − 0.632·10-s − 16-s − 0.727·17-s − 1.88·18-s − 0.447·20-s − 4/5·25-s − 1.67·29-s − 1.41·32-s − 1.02·34-s − 4/3·36-s + 0.821·37-s − 0.468·41-s + 0.596·45-s + 4/7·49-s − 1.13·50-s − 1.37·53-s − 2.36·58-s − 0.256·61-s − 64-s − 0.727·68-s − 0.819·73-s + 1.16·74-s + 0.447·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(115600\)    =    \(2^{4} \cdot 5^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(7.37075\)
Root analytic conductor: \(1.64769\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 115600,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 + T + p T^{2} \)
17$C_2$ \( 1 + 3 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.11.a_u
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
19$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.19.a_h
23$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.23.a_ae
29$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.j_da
31$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.31.a_u
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.af_by
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.d_de
43$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.43.a_f
47$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.47.a_af
53$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.k_el
59$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.59.a_an
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.c_es
67$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.67.a_dm
71$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.71.a_abo
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.73.h_dy
79$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.79.a_dc
83$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.83.a_f
89$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.89.au_ji
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.97.az_ms
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.193935989085420975941120096272, −8.805685297379233901664350422678, −8.131377871062503794316869291376, −7.70675236970571404655007913203, −7.15379777788504115449382910047, −6.42752587247229252086038630023, −6.05132648058618191089493812622, −5.62308983832467948153035251042, −5.02757187557242489855967810899, −4.50988870220065703957496002423, −3.81466132792119821996802419342, −3.41119730433172268480983878374, −2.67982700807126108932538437741, −2.00396258952617676420942139258, 0, 2.00396258952617676420942139258, 2.67982700807126108932538437741, 3.41119730433172268480983878374, 3.81466132792119821996802419342, 4.50988870220065703957496002423, 5.02757187557242489855967810899, 5.62308983832467948153035251042, 6.05132648058618191089493812622, 6.42752587247229252086038630023, 7.15379777788504115449382910047, 7.70675236970571404655007913203, 8.131377871062503794316869291376, 8.805685297379233901664350422678, 9.193935989085420975941120096272

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