Properties

Label 4-2850e2-1.1-c1e2-0-2
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 + 16-s + 2·19-s − 20·29-s + 36-s + 4·41-s − 2·49-s − 16·59-s + 12·61-s − 64-s − 2·76-s + 81-s − 20·89-s − 28·101-s + 4·109-s + 20·116-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1/4·16-s + 0.458·19-s − 3.71·29-s + 1/6·36-s + 0.624·41-s − 2/7·49-s − 2.08·59-s + 1.53·61-s − 1/8·64-s − 0.229·76-s + 1/9·81-s − 2.11·89-s − 2.78·101-s + 0.383·109-s + 1.85·116-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-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

\(L(1)\) \(\approx\) \(0.6239549826\)
\(L(\frac12)\) \(\approx\) \(0.6239549826\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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

−9.069886398799138107344701630448, −8.626054588104362840655806160689, −8.234460279125897784171915564557, −7.70418375914744222843625633973, −7.59884664048498673843922003330, −7.24773382618718142579432258969, −6.57608615012675359818066504997, −6.42411380058286349477381271832, −5.58723367337564041232111332853, −5.58272876096380122791336356599, −5.31576124936702369664580313342, −4.67305116665948802341889046723, −4.02043861735270127049854934749, −3.99840375652004299268911124452, −3.39778997460266987656486339706, −2.91580271732660290936732476790, −2.38323186119880101096785166899, −1.72788069589953960569378938951, −1.29904763236614849315474992908, −0.25887487383778662700780252316, 0.25887487383778662700780252316, 1.29904763236614849315474992908, 1.72788069589953960569378938951, 2.38323186119880101096785166899, 2.91580271732660290936732476790, 3.39778997460266987656486339706, 3.99840375652004299268911124452, 4.02043861735270127049854934749, 4.67305116665948802341889046723, 5.31576124936702369664580313342, 5.58272876096380122791336356599, 5.58723367337564041232111332853, 6.42411380058286349477381271832, 6.57608615012675359818066504997, 7.24773382618718142579432258969, 7.59884664048498673843922003330, 7.70418375914744222843625633973, 8.234460279125897784171915564557, 8.626054588104362840655806160689, 9.069886398799138107344701630448

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