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Properties

Label 4-640332-1.1-c1e2-0-18
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 + 6·5-s + 9-s − 12-s − 6·15-s + 16-s + 6·20-s + 12·23-s + 18·25-s − 27-s + 4·31-s + 36-s + 6·45-s − 48-s − 49-s − 18·53-s + 12·59-s − 6·60-s + 64-s − 12·67-s − 12·69-s − 12·71-s − 18·75-s + 6·80-s + 81-s − 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 2.68·5-s + 1/3·9-s − 0.288·12-s − 1.54·15-s + 1/4·16-s + 1.34·20-s + 2.50·23-s + 18/5·25-s − 0.192·27-s + 0.718·31-s + 1/6·36-s + 0.894·45-s − 0.144·48-s − 1/7·49-s − 2.47·53-s + 1.56·59-s − 0.774·60-s + 1/8·64-s − 1.46·67-s − 1.44·69-s − 1.42·71-s − 2.07·75-s + 0.670·80-s + 1/9·81-s − 0.635·89-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

\(L(1)\) \(\approx\) \(3.582631316\)
\(L(\frac12)\) \(\approx\) \(3.582631316\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
7$C_2$ \( 1 + T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.5.ag_s
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ae_be
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.41.a_cc
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.a_by
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.s_gw
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.61.a_ba
67$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.m_fe
71$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.m_gs
73$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.73.a_aw
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.83.a_g
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.g_gg
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.aq_io
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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

−8.588306284228993322910019079566, −7.80193958212726598200307689555, −7.32183660765912911824542697307, −6.74699709884367190533287768720, −6.51536339758611830456646740412, −6.11827065056679955088934441522, −5.65629265381439104833215299913, −5.29081807796721877713053891305, −4.88049926942091769129528924095, −4.35735486353199090873007604186, −3.19902438498418078923442491301, −2.88983225399967859303554755324, −2.21149254138280714447631529000, −1.60043229560927267257229358724, −1.09490259334852374077037147380, 1.09490259334852374077037147380, 1.60043229560927267257229358724, 2.21149254138280714447631529000, 2.88983225399967859303554755324, 3.19902438498418078923442491301, 4.35735486353199090873007604186, 4.88049926942091769129528924095, 5.29081807796721877713053891305, 5.65629265381439104833215299913, 6.11827065056679955088934441522, 6.51536339758611830456646740412, 6.74699709884367190533287768720, 7.32183660765912911824542697307, 7.80193958212726598200307689555, 8.588306284228993322910019079566

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