Properties

Label 4-2850e2-1.1-c1e2-0-15
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
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 − 9-s + 12·11-s + 16-s − 2·19-s + 16·31-s + 36-s − 24·41-s − 12·44-s + 10·49-s + 24·59-s + 4·61-s − 64-s + 2·76-s − 16·79-s + 81-s + 24·89-s − 12·99-s + 12·101-s − 4·109-s + 86·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 3.61·11-s + 1/4·16-s − 0.458·19-s + 2.87·31-s + 1/6·36-s − 3.74·41-s − 1.80·44-s + 10/7·49-s + 3.12·59-s + 0.512·61-s − 1/8·64-s + 0.229·76-s − 1.80·79-s + 1/9·81-s + 2.54·89-s − 1.20·99-s + 1.19·101-s − 0.383·109-s + 7.81·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.424956993\)
\(L(\frac12)\) \(\approx\) \(3.424956993\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.838180755625629281954261427003, −8.633953515486354492096208844947, −8.279821640476004608263266026263, −8.245815574838388641102914317399, −7.12005626896697026260354829827, −7.10038595559102844680560837326, −6.67944941443376805957723470291, −6.29231337625413764445201314182, −6.17319985475513657134332534072, −5.47148481491704216726392496216, −5.10832060079206385189879465539, −4.38784166328517770241147396801, −4.34385956054244529738492942770, −3.89444916056887763846471219301, −3.33221020521897279673932139690, −3.18121582658307778536870310425, −2.15785556659230915868782397180, −1.81211727436000610200546691753, −1.03701669191881756260791499578, −0.75488886847315970442282072106, 0.75488886847315970442282072106, 1.03701669191881756260791499578, 1.81211727436000610200546691753, 2.15785556659230915868782397180, 3.18121582658307778536870310425, 3.33221020521897279673932139690, 3.89444916056887763846471219301, 4.34385956054244529738492942770, 4.38784166328517770241147396801, 5.10832060079206385189879465539, 5.47148481491704216726392496216, 6.17319985475513657134332534072, 6.29231337625413764445201314182, 6.67944941443376805957723470291, 7.10038595559102844680560837326, 7.12005626896697026260354829827, 8.245815574838388641102914317399, 8.279821640476004608263266026263, 8.633953515486354492096208844947, 8.838180755625629281954261427003

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