Properties

Label 4-2450e2-1.1-c1e2-0-10
Degree $4$
Conductor $6002500$
Sign $1$
Analytic cond. $382.724$
Root an. cond. $4.42304$
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  − 4-s + 5·9-s − 6·11-s + 16-s + 10·19-s − 4·31-s − 5·36-s + 6·41-s + 6·44-s − 4·61-s − 64-s + 24·71-s − 10·76-s + 20·79-s + 16·81-s + 30·89-s − 30·99-s + 36·101-s + 20·109-s + 5·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s + 5/3·9-s − 1.80·11-s + 1/4·16-s + 2.29·19-s − 0.718·31-s − 5/6·36-s + 0.937·41-s + 0.904·44-s − 0.512·61-s − 1/8·64-s + 2.84·71-s − 1.14·76-s + 2.25·79-s + 16/9·81-s + 3.17·89-s − 3.01·99-s + 3.58·101-s + 1.91·109-s + 5/11·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6002500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(382.724\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6002500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.615679209\)
\(L(\frac12)\) \(\approx\) \(2.615679209\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.210897878895161822577777184573, −8.945892250575963369894923740477, −8.111053233997072655314004902403, −7.931341697641231005226120585215, −7.51768774789389237729426652336, −7.49428889006010056802365393282, −6.97561977052450077694041106513, −6.39288274303282700210173489811, −6.06239305749795240628384771694, −5.30872003303464996801529057959, −5.24288503909713039923766936655, −4.86161418702765497160292727101, −4.48590962523561887033755428260, −3.79460407379841790905164282584, −3.44816921109119400284687205159, −3.11660283789478574423296050837, −2.21502812324535321828730172815, −2.04424886370320975969878846508, −1.03216917778632698643065556762, −0.67237267760842111817178513817, 0.67237267760842111817178513817, 1.03216917778632698643065556762, 2.04424886370320975969878846508, 2.21502812324535321828730172815, 3.11660283789478574423296050837, 3.44816921109119400284687205159, 3.79460407379841790905164282584, 4.48590962523561887033755428260, 4.86161418702765497160292727101, 5.24288503909713039923766936655, 5.30872003303464996801529057959, 6.06239305749795240628384771694, 6.39288274303282700210173489811, 6.97561977052450077694041106513, 7.49428889006010056802365393282, 7.51768774789389237729426652336, 7.931341697641231005226120585215, 8.111053233997072655314004902403, 8.945892250575963369894923740477, 9.210897878895161822577777184573

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