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Properties

Label 4-800e2-1.1-c1e2-0-3
Degree $4$
Conductor $640000$
Sign $1$
Analytic cond. $40.8069$
Root an. cond. $2.52745$
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  − 5·9-s − 8·31-s − 6·41-s − 14·49-s + 24·71-s + 16·79-s + 16·81-s − 6·89-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 5/3·9-s − 1.43·31-s − 0.937·41-s − 2·49-s + 2.84·71-s + 1.80·79-s + 16/9·81-s − 0.635·89-s + 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640000\)    =    \(2^{10} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(40.8069\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 640000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.075799996\)
\(L(\frac12)\) \(\approx\) \(1.075799996\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.11.a_at
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.17.a_bf
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.47.a_bu
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.67.a_dh
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 + 71 T^{2} + p^{2} T^{4} \) 2.73.a_ct
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.a_ach
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.89.g_hf
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−8.428486400747804793151703530000, −7.915327623218048963038712631796, −7.69226238293556567279711329602, −6.81912952132796605113592459464, −6.66925180190664287671492954194, −6.05906706107540846561151799389, −5.58899525120293222298727729519, −5.18496433514385854159986925342, −4.81215981119790443785594662350, −3.99182667831283863365527453643, −3.39596266268224950048168817990, −3.11777778252421033443591535343, −2.30075939793565868913691142544, −1.76584322210679783680166776157, −0.50214320024975548831087623862, 0.50214320024975548831087623862, 1.76584322210679783680166776157, 2.30075939793565868913691142544, 3.11777778252421033443591535343, 3.39596266268224950048168817990, 3.99182667831283863365527453643, 4.81215981119790443785594662350, 5.18496433514385854159986925342, 5.58899525120293222298727729519, 6.05906706107540846561151799389, 6.66925180190664287671492954194, 6.81912952132796605113592459464, 7.69226238293556567279711329602, 7.915327623218048963038712631796, 8.428486400747804793151703530000

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