Properties

Label 4-4214-1.1-c1e2-0-0
Degree $4$
Conductor $4214$
Sign $1$
Analytic cond. $0.268688$
Root an. cond. $0.719966$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 7-s + 8-s + 4·9-s + 6·11-s + 14-s + 3·16-s − 4·18-s − 6·22-s − 9·23-s − 25-s + 28-s − 3·32-s − 4·36-s − 14·37-s + 9·43-s − 6·44-s + 9·46-s − 6·49-s + 50-s − 12·53-s − 56-s − 4·63-s − 5·64-s + 10·67-s + 21·71-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.377·7-s + 0.353·8-s + 4/3·9-s + 1.80·11-s + 0.267·14-s + 3/4·16-s − 0.942·18-s − 1.27·22-s − 1.87·23-s − 1/5·25-s + 0.188·28-s − 0.530·32-s − 2/3·36-s − 2.30·37-s + 1.37·43-s − 0.904·44-s + 1.32·46-s − 6/7·49-s + 0.141·50-s − 1.64·53-s − 0.133·56-s − 0.503·63-s − 5/8·64-s + 1.22·67-s + 2.49·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4214 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4214 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4214\)    =    \(2 \cdot 7^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(0.268688\)
Root analytic conductor: \(0.719966\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4214} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4214,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5412741631\)
\(L(\frac12)\) \(\approx\) \(0.5412741631\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
7$C_2$ \( 1 + T + p T^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 8 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \)
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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

−12.40992743119000219830426930703, −12.02493042028340985141242906119, −11.19494436916396067256088003266, −10.41052341605502377252150880437, −9.844829833303857857340526608476, −9.505931410169624923239388157236, −8.994621533683078515979595306289, −8.227068586557118849738869505629, −7.62445480258900598893734797038, −6.71059934653250726444651732388, −6.31644324564184210493440834356, −5.19765492894545924172017700826, −4.07724382311546908680728359391, −3.71331134171430874660508363364, −1.61452194687764831160312520057, 1.61452194687764831160312520057, 3.71331134171430874660508363364, 4.07724382311546908680728359391, 5.19765492894545924172017700826, 6.31644324564184210493440834356, 6.71059934653250726444651732388, 7.62445480258900598893734797038, 8.227068586557118849738869505629, 8.994621533683078515979595306289, 9.505931410169624923239388157236, 9.844829833303857857340526608476, 10.41052341605502377252150880437, 11.19494436916396067256088003266, 12.02493042028340985141242906119, 12.40992743119000219830426930703

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