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Properties

Label 4-1800e2-1.1-c1e2-0-25
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 + 12·13-s − 4·16-s − 24·26-s − 4·31-s + 8·32-s − 4·37-s + 8·41-s + 8·43-s − 2·49-s + 24·52-s + 20·53-s + 8·62-s − 8·64-s + 24·67-s + 16·71-s + 8·74-s + 4·79-s − 16·82-s − 16·86-s + 8·89-s + 4·98-s − 40·106-s − 40·107-s + 18·121-s − 8·124-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 3.32·13-s − 16-s − 4.70·26-s − 0.718·31-s + 1.41·32-s − 0.657·37-s + 1.24·41-s + 1.21·43-s − 2/7·49-s + 3.32·52-s + 2.74·53-s + 1.01·62-s − 64-s + 2.93·67-s + 1.89·71-s + 0.929·74-s + 0.450·79-s − 1.76·82-s − 1.72·86-s + 0.847·89-s + 0.404·98-s − 3.88·106-s − 3.86·107-s + 1.63·121-s − 0.718·124-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

\(L(1)\) \(\approx\) \(1.621197324\)
\(L(\frac12)\) \(\approx\) \(1.621197324\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
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.e_co
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.41.ai_du
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.47.a_ada
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.59.a_aek
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.61.a_aec
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.79.ae_gg
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.89.ai_hm
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

−9.193604476428262694927115996249, −9.156869977634684961611498147786, −8.706311034544052217568415576446, −8.362372008007936551860659670815, −7.962545256253017799483039785862, −7.87684925850585019163152786079, −7.01254489591778499828421736450, −6.86285963184122265869921684291, −6.43748728406915112167096259303, −5.97534387240982335701887714394, −5.49380266067131011559389839433, −5.26225883113365586509648836581, −4.27733583957108090089648976129, −3.93152832254805488652289088242, −3.73290851556379399137999713601, −3.02941537680886862058591919431, −2.24047843681278126090859570014, −1.82823276316141233883811032023, −0.903121479822252690218943990070, −0.885674971016098093782675370385, 0.885674971016098093782675370385, 0.903121479822252690218943990070, 1.82823276316141233883811032023, 2.24047843681278126090859570014, 3.02941537680886862058591919431, 3.73290851556379399137999713601, 3.93152832254805488652289088242, 4.27733583957108090089648976129, 5.26225883113365586509648836581, 5.49380266067131011559389839433, 5.97534387240982335701887714394, 6.43748728406915112167096259303, 6.86285963184122265869921684291, 7.01254489591778499828421736450, 7.87684925850585019163152786079, 7.962545256253017799483039785862, 8.362372008007936551860659670815, 8.706311034544052217568415576446, 9.156869977634684961611498147786, 9.193604476428262694927115996249

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