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Properties

Label 4-1800e2-1.1-c1e2-0-20
Degree $4$
Conductor $3240000$
Sign $1$
Analytic cond. $206.585$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 8·13-s − 4·16-s + 16·26-s + 4·31-s − 8·32-s + 16·37-s − 4·41-s + 8·43-s + 10·49-s + 16·52-s + 12·53-s + 8·62-s − 8·64-s − 24·67-s − 24·71-s + 32·74-s − 20·79-s − 8·82-s + 32·83-s + 16·86-s − 20·89-s + 20·98-s + 24·106-s + 24·107-s + 22·121-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 2.21·13-s − 16-s + 3.13·26-s + 0.718·31-s − 1.41·32-s + 2.63·37-s − 0.624·41-s + 1.21·43-s + 10/7·49-s + 2.21·52-s + 1.64·53-s + 1.01·62-s − 64-s − 2.93·67-s − 2.84·71-s + 3.71·74-s − 2.25·79-s − 0.883·82-s + 3.51·83-s + 1.72·86-s − 2.11·89-s + 2.02·98-s + 2.33·106-s + 2.32·107-s + 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3240000\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(206.585\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3240000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.972333911\)
\(L(\frac12)\) \(\approx\) \(5.972333911\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.83.abg_qg
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−9.314507200618002250533247451487, −9.030282948693486057450184829038, −8.591375150705488739242755706255, −8.503424704837594372126250735292, −7.77811424463970224642902418798, −7.31786813172050784231860706904, −7.13288526493330359549415890384, −6.29493602484131702060044460157, −6.16110717016107376368279304677, −5.79731905573169127111856520655, −5.68418036403554140054220913322, −4.80276296320580186913037152092, −4.48094235981944098067566370914, −4.03921192762374973116087921436, −3.85298257708795142978632907277, −3.00727653981535997344366122134, −2.93070813281972060609281706698, −2.20113947149360345231982950921, −1.40456347650071723392420288314, −0.77252469056645028172796100366, 0.77252469056645028172796100366, 1.40456347650071723392420288314, 2.20113947149360345231982950921, 2.93070813281972060609281706698, 3.00727653981535997344366122134, 3.85298257708795142978632907277, 4.03921192762374973116087921436, 4.48094235981944098067566370914, 4.80276296320580186913037152092, 5.68418036403554140054220913322, 5.79731905573169127111856520655, 6.16110717016107376368279304677, 6.29493602484131702060044460157, 7.13288526493330359549415890384, 7.31786813172050784231860706904, 7.77811424463970224642902418798, 8.503424704837594372126250735292, 8.591375150705488739242755706255, 9.030282948693486057450184829038, 9.314507200618002250533247451487

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