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Properties

Label 4-640332-1.1-c1e2-0-2
Degree $4$
Conductor $640332$
Sign $1$
Analytic cond. $40.8281$
Root an. cond. $2.52778$
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 − 4-s + 4·7-s + 9-s + 12-s − 3·13-s + 16-s + 19-s − 4·21-s − 4·25-s − 27-s − 4·28-s + 31-s − 36-s + 12·37-s + 3·39-s + 43-s − 48-s + 9·49-s + 3·52-s − 57-s + 4·61-s + 4·63-s − 64-s + 67-s + 11·73-s + 4·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 1.51·7-s + 1/3·9-s + 0.288·12-s − 0.832·13-s + 1/4·16-s + 0.229·19-s − 0.872·21-s − 4/5·25-s − 0.192·27-s − 0.755·28-s + 0.179·31-s − 1/6·36-s + 1.97·37-s + 0.480·39-s + 0.152·43-s − 0.144·48-s + 9/7·49-s + 0.416·52-s − 0.132·57-s + 0.512·61-s + 0.503·63-s − 1/8·64-s + 0.122·67-s + 1.28·73-s + 0.461·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640332\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(40.8281\)
Root analytic conductor: \(2.52778\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 640332,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.538270921\)
\(L(\frac12)\) \(\approx\) \(1.538270921\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 + T \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
11$C_2$ \( 1 + T^{2} \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
13$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.d_i
17$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.17.a_ae
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ab_bk
23$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.23.a_bf
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ab_g
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.37.am_dh
41$C_2^2$ \( 1 - 72 T^{2} + p^{2} T^{4} \) 2.41.a_acu
43$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.ab_o
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.53.a_ak
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.59.a_w
61$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.61.ae_es
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.67.ab_es
71$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.71.a_h
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.73.al_ea
79$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.al_gc
83$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.83.a_aj
89$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.89.a_du
97$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.y_mw
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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

−8.216096205934668118763810471928, −7.924184039458965307970266991815, −7.60330061268231049108577457540, −7.13052660499546511716575866953, −6.50498030010084657722846659176, −6.03480911341987142463494127345, −5.48007760667757149203141497643, −5.06392971988162219875088408256, −4.78863245839722340099350144101, −4.12341316941475222245974016781, −3.89454281674751665379636023548, −2.84518681226484191776073213440, −2.25566872200255148980860736710, −1.52720044618230074469814272479, −0.68641521185623661782087662897, 0.68641521185623661782087662897, 1.52720044618230074469814272479, 2.25566872200255148980860736710, 2.84518681226484191776073213440, 3.89454281674751665379636023548, 4.12341316941475222245974016781, 4.78863245839722340099350144101, 5.06392971988162219875088408256, 5.48007760667757149203141497643, 6.03480911341987142463494127345, 6.50498030010084657722846659176, 7.13052660499546511716575866953, 7.60330061268231049108577457540, 7.924184039458965307970266991815, 8.216096205934668118763810471928

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