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Properties

Label 4-30e4-1.1-c1e2-0-12
Degree $4$
Conductor $810000$
Sign $-1$
Analytic cond. $51.6463$
Root an. cond. $2.68077$
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  − 2·2-s + 2·4-s + 2·13-s − 4·16-s − 4·17-s − 4·26-s − 20·29-s + 8·32-s + 8·34-s + 4·37-s + 16·41-s − 5·49-s + 4·52-s + 8·53-s + 40·58-s + 14·61-s − 8·64-s − 8·68-s − 28·73-s − 8·74-s − 32·82-s + 34·97-s + 10·98-s − 24·101-s − 16·106-s + 10·109-s + 8·113-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.554·13-s − 16-s − 0.970·17-s − 0.784·26-s − 3.71·29-s + 1.41·32-s + 1.37·34-s + 0.657·37-s + 2.49·41-s − 5/7·49-s + 0.554·52-s + 1.09·53-s + 5.25·58-s + 1.79·61-s − 64-s − 0.970·68-s − 3.27·73-s − 0.929·74-s − 3.53·82-s + 3.45·97-s + 1.01·98-s − 2.38·101-s − 1.55·106-s + 0.957·109-s + 0.752·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(810000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(51.6463\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 810000,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.29.u_gc
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.41.aq_fq
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.a_dh
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.67.a_ev
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.97.abi_sp
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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

−7.965886472126962800720761127653, −7.62852485359829009908403777253, −7.32922657340821485296829402521, −6.96141726621690601574492077524, −6.27130274152454875896945514364, −5.87930303915993667052597404250, −5.48558822518018321489922753080, −4.75727383162943095433406736583, −4.01465426298836833468759021488, −3.98830812112494448268150912948, −3.01327782908855925530719016170, −2.23186561512245260821665501913, −1.87603850679094242426395746447, −0.985333224458569944413443756671, 0, 0.985333224458569944413443756671, 1.87603850679094242426395746447, 2.23186561512245260821665501913, 3.01327782908855925530719016170, 3.98830812112494448268150912948, 4.01465426298836833468759021488, 4.75727383162943095433406736583, 5.48558822518018321489922753080, 5.87930303915993667052597404250, 6.27130274152454875896945514364, 6.96141726621690601574492077524, 7.32922657340821485296829402521, 7.62852485359829009908403777253, 7.965886472126962800720761127653

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