Properties

Label 4-837e2-1.1-c1e2-0-1
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 − 5·13-s + 12·16-s + 7·19-s + 5·25-s + 20·28-s − 4·31-s − 11·37-s + 13·43-s + 7·49-s + 20·52-s − 26·61-s − 32·64-s − 11·67-s + 10·73-s − 28·76-s + 13·79-s + 25·91-s + 28·97-s − 20·100-s + 13·103-s − 38·109-s − 60·112-s + 11·121-s + 16·124-s + 127-s + ⋯
L(s)  = 1  − 2·4-s − 1.88·7-s − 1.38·13-s + 3·16-s + 1.60·19-s + 25-s + 3.77·28-s − 0.718·31-s − 1.80·37-s + 1.98·43-s + 49-s + 2.77·52-s − 3.32·61-s − 4·64-s − 1.34·67-s + 1.17·73-s − 3.21·76-s + 1.46·79-s + 2.62·91-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 + ⋯

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

\(L(1)\) \(\approx\) \(0.3162673613\)
\(L(\frac12)\) \(\approx\) \(0.3162673613\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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

−10.23911253410641386096303162398, −9.767153770172247781825605372214, −9.579309307953439720599843668315, −9.154639422558579953632851551919, −9.009284841775359080702891587317, −8.557917337072213016946636717322, −7.65890224982241180422039573049, −7.55287597006040447031548057262, −7.18971021275121560053963647773, −6.32762759075367426521689325823, −6.09151175569612241090636988926, −5.43965434149948293451727669768, −4.90120065966546164274104935045, −4.85035423528459677832941700549, −4.01444601200004275990158965301, −3.34661052169739300248281326380, −3.32915063279951232726403372303, −2.57154021486041601091222152967, −1.26681275716438589695621080791, −0.31375434926767570178897708878, 0.31375434926767570178897708878, 1.26681275716438589695621080791, 2.57154021486041601091222152967, 3.32915063279951232726403372303, 3.34661052169739300248281326380, 4.01444601200004275990158965301, 4.85035423528459677832941700549, 4.90120065966546164274104935045, 5.43965434149948293451727669768, 6.09151175569612241090636988926, 6.32762759075367426521689325823, 7.18971021275121560053963647773, 7.55287597006040447031548057262, 7.65890224982241180422039573049, 8.557917337072213016946636717322, 9.009284841775359080702891587317, 9.154639422558579953632851551919, 9.579309307953439720599843668315, 9.767153770172247781825605372214, 10.23911253410641386096303162398

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