↑

Properties

Label 4-260000-1.1-c1e2-0-6
Degree $4$
Conductor $260000$
Sign $-1$
Analytic cond. $16.5778$
Root an. cond. $2.01781$
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-s + 4-s − 8-s − 2·9-s − 3·13-s + 16-s + 2·18-s + 3·26-s − 32-s − 2·36-s + 20·37-s − 12·41-s + 2·49-s − 3·52-s − 12·53-s − 8·61-s + 64-s + 2·72-s − 4·73-s − 20·74-s − 5·81-s + 12·82-s − 4·97-s − 2·98-s + 12·101-s + 3·104-s + 12·106-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s − 0.832·13-s + 1/4·16-s + 0.471·18-s + 0.588·26-s − 0.176·32-s − 1/3·36-s + 3.28·37-s − 1.87·41-s + 2/7·49-s − 0.416·52-s − 1.64·53-s − 1.02·61-s + 1/8·64-s + 0.235·72-s − 0.468·73-s − 2.32·74-s − 5/9·81-s + 1.32·82-s − 0.406·97-s − 0.202·98-s + 1.19·101-s + 0.294·104-s + 1.16·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(260000\)    =    \(2^{5} \cdot 5^{4} \cdot 13\)
Sign: $-1$
Analytic conductor: \(16.5778\)
Root analytic conductor: \(2.01781\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 260000,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
5 \( 1 \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.43.a_bi
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.67.a_w
71$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.71.a_aba
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.e_cc
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

−8.785807306881286673948764335556, −8.127894199113213770452281445118, −7.79342019470039543486313986763, −7.51768334420833827457211375191, −6.74735337572119730897669452296, −6.39832991660811317676841413369, −5.89331301156813931130797705476, −5.33784225359882312425656865091, −4.71778063106321092665172507836, −4.23315891390327269038768825313, −3.32397427405087055171712695713, −2.81054304357396216223823514236, −2.22582990295811625604875462143, −1.25731677920143884915232085828, 0, 1.25731677920143884915232085828, 2.22582990295811625604875462143, 2.81054304357396216223823514236, 3.32397427405087055171712695713, 4.23315891390327269038768825313, 4.71778063106321092665172507836, 5.33784225359882312425656865091, 5.89331301156813931130797705476, 6.39832991660811317676841413369, 6.74735337572119730897669452296, 7.51768334420833827457211375191, 7.79342019470039543486313986763, 8.127894199113213770452281445118, 8.785807306881286673948764335556

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