Properties

Label 4-2800e2-1.1-c1e2-0-15
Degree $4$
Conductor $7840000$
Sign $1$
Analytic cond. $499.885$
Root an. cond. $4.72843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·9-s − 2·11-s − 4·19-s − 2·29-s + 4·31-s + 24·41-s − 49-s − 20·59-s + 8·61-s + 6·71-s − 18·79-s + 27·81-s + 12·89-s − 12·99-s + 24·101-s − 10·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  + 2·9-s − 0.603·11-s − 0.917·19-s − 0.371·29-s + 0.718·31-s + 3.74·41-s − 1/7·49-s − 2.60·59-s + 1.02·61-s + 0.712·71-s − 2.02·79-s + 3·81-s + 1.27·89-s − 1.20·99-s + 2.38·101-s − 0.957·109-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 + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7840000\)    =    \(2^{8} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(499.885\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7840000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.994071556\)
\(L(\frac12)\) \(\approx\) \(2.994071556\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 133 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−8.987580220894569765943456026911, −8.767676850893497549172386042600, −7.983756260835750712005208834580, −7.74662939867041886646242846993, −7.64626743882928180275023609940, −7.19080763430036140943586240738, −6.56093804928038261803541032505, −6.53115139993931420625363737273, −5.94224177464664418584149948307, −5.57794553737165345132575280987, −5.02738150763096456837183848231, −4.51084993773464118217159838039, −4.20888597714119995921158889225, −4.15525470121499606203466314814, −3.32876956629738001565344348671, −2.88356868792995965327967952381, −2.22956998807617659933941189387, −1.90652841912408102382557785599, −1.18700357345533808297043422356, −0.61307819382894208145769915765, 0.61307819382894208145769915765, 1.18700357345533808297043422356, 1.90652841912408102382557785599, 2.22956998807617659933941189387, 2.88356868792995965327967952381, 3.32876956629738001565344348671, 4.15525470121499606203466314814, 4.20888597714119995921158889225, 4.51084993773464118217159838039, 5.02738150763096456837183848231, 5.57794553737165345132575280987, 5.94224177464664418584149948307, 6.53115139993931420625363737273, 6.56093804928038261803541032505, 7.19080763430036140943586240738, 7.64626743882928180275023609940, 7.74662939867041886646242846993, 7.983756260835750712005208834580, 8.767676850893497549172386042600, 8.987580220894569765943456026911

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