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Properties

Label 4-640332-1.1-c1e2-0-16
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 4·11-s + 12-s − 16-s + 2·17-s + 18-s − 4·22-s + 3·24-s + 2·25-s − 27-s − 2·29-s − 4·31-s + 5·32-s + 4·33-s + 2·34-s − 36-s + 4·37-s + 2·41-s + 4·44-s + 48-s − 49-s + 2·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.852·22-s + 0.612·24-s + 2/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.883·32-s + 0.696·33-s + 0.342·34-s − 1/6·36-s + 0.657·37-s + 0.312·41-s + 0.603·44-s + 0.144·48-s − 1/7·49-s + 0.282·50-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: \(1\)
Selberg data: \((4,\ 640332,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 + T \)
7$C_2$ \( 1 + T^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ac_bi
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.c_k
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.e_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ac_de
43$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.43.a_k
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.47.a_ac
53$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.53.a_o
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.a_ada
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.71.a_acc
73$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.73.a_aw
79$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.79.a_abu
83$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) 2.83.abc_nu
89$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.89.a_adu
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−8.042386013242057191759281068005, −7.73193875474790559151689143913, −7.24357262770561757986879026216, −6.67085945837001293384792184240, −6.12078675810264991693813681753, −5.79087706991701204105707454300, −5.24980509079856172672111376176, −5.00061400381632893653341455831, −4.50681549772187313325034835961, −3.86065431032475177551296901352, −3.42631802601421961397147315496, −2.76690929875218979376795267491, −2.15915022974662070276954696654, −1.02819367310663180287905065154, 0, 1.02819367310663180287905065154, 2.15915022974662070276954696654, 2.76690929875218979376795267491, 3.42631802601421961397147315496, 3.86065431032475177551296901352, 4.50681549772187313325034835961, 5.00061400381632893653341455831, 5.24980509079856172672111376176, 5.79087706991701204105707454300, 6.12078675810264991693813681753, 6.67085945837001293384792184240, 7.24357262770561757986879026216, 7.73193875474790559151689143913, 8.042386013242057191759281068005

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