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Properties

Label 4-1800e2-1.1-c1e2-0-7
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.m_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.ae_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.i_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.u_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.y_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

−9.516061643001699773332940007190, −9.083036447714271327438519503864, −8.910168511182405858955809737234, −8.019125174631113372499180427745, −7.73346092739650315909500991897, −7.47754021120688015074979337510, −7.00120183010161195083190607968, −6.68954276846004909212028006434, −6.05789323795730387692952724101, −5.87449035758814042716370077982, −5.21110462052087592162617998187, −4.79898371834878162805761280085, −4.69931443500905577394490657270, −4.37022307301714551849780774206, −3.41946103624942570234263870168, −3.30633519314885203279983963727, −2.56034412592333479285124990854, −2.31646103326752231060124674407, −1.70560832776912104271242698075, −0.34238756816583435796523064631, 0.34238756816583435796523064631, 1.70560832776912104271242698075, 2.31646103326752231060124674407, 2.56034412592333479285124990854, 3.30633519314885203279983963727, 3.41946103624942570234263870168, 4.37022307301714551849780774206, 4.69931443500905577394490657270, 4.79898371834878162805761280085, 5.21110462052087592162617998187, 5.87449035758814042716370077982, 6.05789323795730387692952724101, 6.68954276846004909212028006434, 7.00120183010161195083190607968, 7.47754021120688015074979337510, 7.73346092739650315909500991897, 8.019125174631113372499180427745, 8.910168511182405858955809737234, 9.083036447714271327438519503864, 9.516061643001699773332940007190

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