Properties

Label 4-837e2-1.1-c1e2-0-7
Degree $4$
Conductor $700569$
Sign $1$
Analytic cond. $44.6688$
Root an. cond. $2.58524$
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  + 4·4-s − 5·7-s + 9·13-s + 12·16-s − 7·19-s − 5·25-s − 20·28-s + 4·31-s + 9·37-s + 3·43-s + 7·49-s + 36·52-s + 32·64-s + 11·67-s + 24·73-s − 28·76-s − 21·79-s − 45·91-s + 28·97-s − 20·100-s − 13·103-s − 38·109-s − 60·112-s + 11·121-s + 16·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 2·4-s − 1.88·7-s + 2.49·13-s + 3·16-s − 1.60·19-s − 25-s − 3.77·28-s + 0.718·31-s + 1.47·37-s + 0.457·43-s + 49-s + 4.99·52-s + 4·64-s + 1.34·67-s + 2.80·73-s − 3.21·76-s − 2.36·79-s − 4.71·91-s + 2.84·97-s − 2·100-s − 1.28·103-s − 3.63·109-s − 5.66·112-s + 121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(700569\)    =    \(3^{6} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(44.6688\)
Root analytic conductor: \(2.58524\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{837} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 700569,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.049971979\)
\(L(\frac12)\) \(\approx\) \(3.049971979\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
31$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−10.38906827642735866738781604118, −10.19946170759739813632001890115, −9.573793682984379580503469082135, −9.366647916484757163315323806657, −8.571603791360317168259616792372, −8.212228305222830744093237394997, −7.966982487468075399162842259388, −7.26491832133683274984688223973, −6.63472160441853148145316048406, −6.60186605864454167863520068670, −6.08606564867199397062043696805, −6.03316770637505305075552413659, −5.48253682978708830472600486482, −4.35630763141934893195528245498, −3.65328169563820756027401966491, −3.62708941106990905197283893945, −2.86540413891020270540426299325, −2.41555362828993481173566442498, −1.71551518566456141501732307859, −0.861651706826499882621914749961, 0.861651706826499882621914749961, 1.71551518566456141501732307859, 2.41555362828993481173566442498, 2.86540413891020270540426299325, 3.62708941106990905197283893945, 3.65328169563820756027401966491, 4.35630763141934893195528245498, 5.48253682978708830472600486482, 6.03316770637505305075552413659, 6.08606564867199397062043696805, 6.60186605864454167863520068670, 6.63472160441853148145316048406, 7.26491832133683274984688223973, 7.966982487468075399162842259388, 8.212228305222830744093237394997, 8.571603791360317168259616792372, 9.366647916484757163315323806657, 9.573793682984379580503469082135, 10.19946170759739813632001890115, 10.38906827642735866738781604118

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