Properties

Label 4-2450e2-1.1-c1e2-0-11
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·9-s + 6·11-s + 16-s + 2·19-s + 12·29-s − 8·31-s − 2·36-s + 6·41-s − 6·44-s − 8·61-s − 64-s + 24·71-s − 2·76-s + 20·79-s − 5·81-s − 12·89-s + 12·99-s − 24·101-s + 8·109-s − 12·116-s + 5·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s + 2/3·9-s + 1.80·11-s + 1/4·16-s + 0.458·19-s + 2.22·29-s − 1.43·31-s − 1/3·36-s + 0.937·41-s − 0.904·44-s − 1.02·61-s − 1/8·64-s + 2.84·71-s − 0.229·76-s + 2.25·79-s − 5/9·81-s − 1.27·89-s + 1.20·99-s − 2.38·101-s + 0.766·109-s − 1.11·116-s + 5/11·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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

\(L(1)\) \(\approx\) \(2.975893502\)
\(L(\frac12)\) \(\approx\) \(2.975893502\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
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

−9.353352723544831242184035705918, −8.912792581926207677104669623008, −8.331075190369435722032575783548, −8.043557958972856613391636881429, −7.67497637698218421622427193859, −7.11167948837869358277620588115, −6.78156099532944409876146794047, −6.51258983071990516320112485269, −6.16709069193190434481853565281, −5.48233414928129521784509526972, −5.29061019124048519003663550667, −4.54246182379169351962363756360, −4.45768563292731942658600692863, −3.81933650691979871125135077726, −3.65078736651990229680661268611, −3.01103752860562333010638456714, −2.42217099602311934520845320210, −1.69940574596552653217504142222, −1.21337527918625568825321105999, −0.67337244601238629872055702984, 0.67337244601238629872055702984, 1.21337527918625568825321105999, 1.69940574596552653217504142222, 2.42217099602311934520845320210, 3.01103752860562333010638456714, 3.65078736651990229680661268611, 3.81933650691979871125135077726, 4.45768563292731942658600692863, 4.54246182379169351962363756360, 5.29061019124048519003663550667, 5.48233414928129521784509526972, 6.16709069193190434481853565281, 6.51258983071990516320112485269, 6.78156099532944409876146794047, 7.11167948837869358277620588115, 7.67497637698218421622427193859, 8.043557958972856613391636881429, 8.331075190369435722032575783548, 8.912792581926207677104669623008, 9.353352723544831242184035705918

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