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Properties

Label 4-1440e2-1.1-c1e2-0-12
Degree $4$
Conductor $2073600$
Sign $1$
Analytic cond. $132.214$
Root an. cond. $3.39093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 2·11-s + 17-s − 9·19-s − 25-s + 27-s + 2·33-s − 13·41-s − 3·43-s + 6·49-s + 51-s − 9·57-s + 13·67-s + 14·73-s − 75-s + 81-s − 4·83-s + 9·89-s + 18·97-s + 2·99-s − 20·107-s + 25·113-s − 18·121-s − 13·123-s + 127-s − 3·129-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.242·17-s − 2.06·19-s − 1/5·25-s + 0.192·27-s + 0.348·33-s − 2.03·41-s − 0.457·43-s + 6/7·49-s + 0.140·51-s − 1.19·57-s + 1.58·67-s + 1.63·73-s − 0.115·75-s + 1/9·81-s − 0.439·83-s + 0.953·89-s + 1.82·97-s + 0.201·99-s − 1.93·107-s + 2.35·113-s − 1.63·121-s − 1.17·123-s + 0.0887·127-s − 0.264·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2073600\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(132.214\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2073600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.327690553\)
\(L(\frac12)\) \(\approx\) \(2.327690553\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( 1 - T \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ac_w
13$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.13.a_h
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.ab_w
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.j_ca
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.23.a_abl
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.37.a_f
41$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.n_es
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.d_cy
47$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.47.a_abl
53$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \) 2.53.a_ce
59$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.59.a_ad
61$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.61.a_o
67$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.an_ea
71$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.71.a_h
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.73.ao_hn
79$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \) 2.79.a_ea
83$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.83.e_gn
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.89.aj_gm
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.97.as_ko
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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

−7.82126628145067219209593330318, −7.29668484751249940518922125368, −6.82763642700089929121350372087, −6.54912785014526699084986980746, −6.19058271873125930053577278119, −5.60957446274182471794276843766, −5.06335769921059433732190030637, −4.64951244666879282673773601207, −4.13802114252349875838931327433, −3.65146900650596994231790871573, −3.37506310092476169535610004049, −2.54587863722260339503119939903, −2.07039258377071037076247629063, −1.61258210128332938872577577286, −0.58550777041650015333730529553, 0.58550777041650015333730529553, 1.61258210128332938872577577286, 2.07039258377071037076247629063, 2.54587863722260339503119939903, 3.37506310092476169535610004049, 3.65146900650596994231790871573, 4.13802114252349875838931327433, 4.64951244666879282673773601207, 5.06335769921059433732190030637, 5.60957446274182471794276843766, 6.19058271873125930053577278119, 6.54912785014526699084986980746, 6.82763642700089929121350372087, 7.29668484751249940518922125368, 7.82126628145067219209593330318

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