Properties

Label 4-40e4-1.1-c1e2-0-18
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s + 12·17-s − 12·41-s − 14·49-s − 4·73-s − 5·81-s + 36·89-s − 20·97-s − 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 2/3·9-s + 2.91·17-s − 1.87·41-s − 2·49-s − 0.468·73-s − 5/9·81-s + 3.81·89-s − 2.03·97-s − 3.38·113-s − 1.27·121-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.537997386\)
\(L(\frac12)\) \(\approx\) \(2.537997386\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−9.694416774443088210999715477646, −9.313892019799464381556970285465, −8.851616209863958700260998977023, −8.172631021432919553864616120885, −8.103774700995373890607445658339, −7.60788838758082033607254322932, −7.37550334194041508567538602736, −6.68281195308945160314481488920, −6.50593193379168764596163133492, −5.95307842161694148323001762585, −5.28142301435937642599599931765, −5.26516734667149971724317250711, −4.74599182667682973572692993289, −3.92857836024839309430758945156, −3.77096149761736834639654490818, −2.98866927850474552144855626409, −2.94778991351403219971777831753, −1.65869642098408123024634693077, −1.58120374473771042556784119673, −0.65002429655742763620723412684, 0.65002429655742763620723412684, 1.58120374473771042556784119673, 1.65869642098408123024634693077, 2.94778991351403219971777831753, 2.98866927850474552144855626409, 3.77096149761736834639654490818, 3.92857836024839309430758945156, 4.74599182667682973572692993289, 5.26516734667149971724317250711, 5.28142301435937642599599931765, 5.95307842161694148323001762585, 6.50593193379168764596163133492, 6.68281195308945160314481488920, 7.37550334194041508567538602736, 7.60788838758082033607254322932, 8.103774700995373890607445658339, 8.172631021432919553864616120885, 8.851616209863958700260998977023, 9.313892019799464381556970285465, 9.694416774443088210999715477646

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