Properties

Label 4-2850e2-1.1-c1e2-0-20
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 8·11-s + 16-s + 2·19-s − 12·29-s − 16·31-s + 36-s + 20·41-s + 8·44-s − 2·49-s + 8·59-s − 4·61-s − 64-s − 32·71-s − 2·76-s − 16·79-s + 81-s + 12·89-s + 8·99-s + 12·101-s − 12·109-s + 12·116-s + 26·121-s + 16·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s + 0.458·19-s − 2.22·29-s − 2.87·31-s + 1/6·36-s + 3.12·41-s + 1.20·44-s − 2/7·49-s + 1.04·59-s − 0.512·61-s − 1/8·64-s − 3.79·71-s − 0.229·76-s − 1.80·79-s + 1/9·81-s + 1.27·89-s + 0.804·99-s + 1.19·101-s − 1.14·109-s + 1.11·116-s + 2.36·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-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: \(2\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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

−8.703273392880148594582583378676, −7.998101945442112101097968290688, −7.88700471717932672810990170542, −7.39169658548347539878920690744, −7.38711328891432270106239604100, −6.92792546672046921019481485190, −5.83154338404942666368037420418, −5.82864038938981792931961554854, −5.68057094767375485779222580700, −5.19697352013823300227655925677, −4.74056725615326966757794434412, −4.23028916231228240379328945862, −3.88073497093061971168600235641, −3.12979719336492174869604714731, −3.07851030095443522553505219361, −2.22918500235320419014572217662, −2.05515554963146575592832786288, −1.14966679848440505616067908841, 0, 0, 1.14966679848440505616067908841, 2.05515554963146575592832786288, 2.22918500235320419014572217662, 3.07851030095443522553505219361, 3.12979719336492174869604714731, 3.88073497093061971168600235641, 4.23028916231228240379328945862, 4.74056725615326966757794434412, 5.19697352013823300227655925677, 5.68057094767375485779222580700, 5.82864038938981792931961554854, 5.83154338404942666368037420418, 6.92792546672046921019481485190, 7.38711328891432270106239604100, 7.39169658548347539878920690744, 7.88700471717932672810990170542, 7.998101945442112101097968290688, 8.703273392880148594582583378676

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