Properties

Label 4-731-1.1-c1e2-0-0
Degree $4$
Conductor $731$
Sign $1$
Analytic cond. $0.0466091$
Root an. cond. $0.464641$
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 − 2·7-s + 8-s + 11-s − 5·13-s + 2·14-s − 16-s + 2·17-s + 8·19-s − 22-s + 23-s − 4·25-s + 5·26-s + 2·28-s + 6·29-s − 31-s + 5·32-s − 2·34-s − 10·37-s − 8·38-s + 3·41-s + 3·43-s − 44-s − 46-s − 4·47-s + 4·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.301·11-s − 1.38·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s + 1.83·19-s − 0.213·22-s + 0.208·23-s − 4/5·25-s + 0.980·26-s + 0.377·28-s + 1.11·29-s − 0.179·31-s + 0.883·32-s − 0.342·34-s − 1.64·37-s − 1.29·38-s + 0.468·41-s + 0.457·43-s − 0.150·44-s − 0.147·46-s − 0.583·47-s + 0.565·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $1$
Analytic conductor: \(0.0466091\)
Root analytic conductor: \(0.464641\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{731} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 731,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2985355886\)
\(L(\frac12)\) \(\approx\) \(0.2985355886\)
\(L(\frac{3}{2})\) 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)$
bad17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T - 38 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T - 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 5 T + 10 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 124 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−19.6292549877, −19.4633738049, −18.9119628461, −18.2218017754, −17.6236418664, −17.4091766255, −16.5779861033, −16.0045134328, −15.5073211525, −14.5873206745, −13.9972394337, −13.5450844559, −12.5940116925, −12.0779199048, −11.4656015212, −10.2319654667, −9.78503731135, −9.35255019021, −8.54118909771, −7.58446247912, −6.91477576721, −5.68806603501, −4.67056299587, −3.16443864831, 3.16443864831, 4.67056299587, 5.68806603501, 6.91477576721, 7.58446247912, 8.54118909771, 9.35255019021, 9.78503731135, 10.2319654667, 11.4656015212, 12.0779199048, 12.5940116925, 13.5450844559, 13.9972394337, 14.5873206745, 15.5073211525, 16.0045134328, 16.5779861033, 17.4091766255, 17.6236418664, 18.2218017754, 18.9119628461, 19.4633738049, 19.6292549877

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