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Properties

Label 4-30e4-1.1-c1e2-0-18
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.c_bb
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_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.e_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.i_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.abc_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.bi_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

−8.045639809832312220968668237407, −7.46678658042041033049343800975, −6.99182605075727565742724929802, −6.66175602749173699922625283942, −5.96391714742476427877734512505, −5.60694328736998842121660037613, −5.31508624669203565794999838972, −4.92213034669592094010306499374, −4.12503217255581132297434015747, −3.78545707329788062312381068333, −3.50877292142339574199218689538, −2.63519340777820009043746148182, −2.27079729708413704097834802658, −1.40715345955998479729965074286, 0, 1.40715345955998479729965074286, 2.27079729708413704097834802658, 2.63519340777820009043746148182, 3.50877292142339574199218689538, 3.78545707329788062312381068333, 4.12503217255581132297434015747, 4.92213034669592094010306499374, 5.31508624669203565794999838972, 5.60694328736998842121660037613, 5.96391714742476427877734512505, 6.66175602749173699922625283942, 6.99182605075727565742724929802, 7.46678658042041033049343800975, 8.045639809832312220968668237407

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