Properties

Label 4-2850e2-1.1-c1e2-0-6
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 + 4·11-s + 16-s + 2·19-s − 16·31-s + 36-s − 16·41-s − 4·44-s + 10·49-s + 4·61-s − 64-s − 16·71-s − 2·76-s + 81-s − 4·99-s + 4·101-s − 20·109-s − 10·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1.20·11-s + 1/4·16-s + 0.458·19-s − 2.87·31-s + 1/6·36-s − 2.49·41-s − 0.603·44-s + 10/7·49-s + 0.512·61-s − 1/8·64-s − 1.89·71-s − 0.229·76-s + 1/9·81-s − 0.402·99-s + 0.398·101-s − 1.91·109-s − 0.909·121-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 + 0.0819·149-s + 0.0813·151-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

\(L(1)\) \(\approx\) \(1.079111608\)
\(L(\frac12)\) \(\approx\) \(1.079111608\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−9.156799685391133279097436904425, −8.680154042014038424836184127659, −8.404032806921054579354080136976, −7.63566712124265102626018713438, −7.62667610961840292057413200458, −7.07669574052699122284242608032, −6.68195493960369995816912778209, −6.42002802373597097689393308297, −5.73851389389688995775799905267, −5.56601505946024522728239876014, −5.10773717995314291651008490744, −4.79431436958230531772701946549, −4.04496712665550083954375539417, −3.79104109000987595875768053949, −3.57744448825122072318256757528, −2.91850913226440531604287501939, −2.37937152391413829639298038725, −1.53386947559893321155787256698, −1.44941456456430841887347479534, −0.33966234015730159263296938501, 0.33966234015730159263296938501, 1.44941456456430841887347479534, 1.53386947559893321155787256698, 2.37937152391413829639298038725, 2.91850913226440531604287501939, 3.57744448825122072318256757528, 3.79104109000987595875768053949, 4.04496712665550083954375539417, 4.79431436958230531772701946549, 5.10773717995314291651008490744, 5.56601505946024522728239876014, 5.73851389389688995775799905267, 6.42002802373597097689393308297, 6.68195493960369995816912778209, 7.07669574052699122284242608032, 7.62667610961840292057413200458, 7.63566712124265102626018713438, 8.404032806921054579354080136976, 8.680154042014038424836184127659, 9.156799685391133279097436904425

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