Properties

Label 4-300e2-1.1-c1e2-0-9
Degree $4$
Conductor $90000$
Sign $-1$
Analytic cond. $5.73847$
Root an. cond. $1.54774$
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  − 9-s − 2·11-s − 6·19-s − 12·29-s − 8·31-s + 4·41-s − 6·49-s + 4·59-s − 8·61-s + 14·71-s + 2·79-s + 81-s − 20·89-s + 2·99-s + 8·101-s − 16·109-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + ⋯
L(s)  = 1  − 1/3·9-s − 0.603·11-s − 1.37·19-s − 2.22·29-s − 1.43·31-s + 0.624·41-s − 6/7·49-s + 0.520·59-s − 1.02·61-s + 1.66·71-s + 0.225·79-s + 1/9·81-s − 2.11·89-s + 0.201·99-s + 0.796·101-s − 1.53·109-s − 1.63·121-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 − 0.307·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(90000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(5.73847\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 90000,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 36 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
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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

−9.422197574188130817763059589262, −8.965053913275388148374816983982, −8.356973850486090935420059188355, −7.957883933574345054701003107581, −7.41998954855005023127010946195, −6.91314094544873394457774872417, −6.30863413441593372512421985066, −5.63139352270634351168527672090, −5.39543344743137463832307303895, −4.55454017653786936444030370390, −3.93635944741692071534317197982, −3.34241924400158333321788587678, −2.41986240838994355096835391492, −1.77673945351887851385720722094, 0, 1.77673945351887851385720722094, 2.41986240838994355096835391492, 3.34241924400158333321788587678, 3.93635944741692071534317197982, 4.55454017653786936444030370390, 5.39543344743137463832307303895, 5.63139352270634351168527672090, 6.30863413441593372512421985066, 6.91314094544873394457774872417, 7.41998954855005023127010946195, 7.957883933574345054701003107581, 8.356973850486090935420059188355, 8.965053913275388148374816983982, 9.422197574188130817763059589262

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