Properties

Label 4-39e4-1.1-c1e2-0-1
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $147.507$
Root an. cond. $3.48500$
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  − 2·2-s + 4-s − 4·11-s + 16-s − 4·17-s + 8·22-s + 8·23-s − 2·25-s − 4·29-s + 8·31-s + 2·32-s + 8·34-s + 4·37-s + 16·41-s + 8·43-s − 4·44-s − 16·46-s − 12·47-s − 6·49-s + 4·50-s + 4·53-s + 8·58-s + 4·59-s + 4·61-s − 16·62-s − 11·64-s − 8·67-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s − 1.20·11-s + 1/4·16-s − 0.970·17-s + 1.70·22-s + 1.66·23-s − 2/5·25-s − 0.742·29-s + 1.43·31-s + 0.353·32-s + 1.37·34-s + 0.657·37-s + 2.49·41-s + 1.21·43-s − 0.603·44-s − 2.35·46-s − 1.75·47-s − 6/7·49-s + 0.565·50-s + 0.549·53-s + 1.05·58-s + 0.520·59-s + 0.512·61-s − 2.03·62-s − 1.37·64-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(147.507\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6002963593\)
\(L(\frac12)\) \(\approx\) \(0.6002963593\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−9.383427021460102407093609518027, −9.353210459151241721163483888227, −8.845823712072260614020394633224, −8.688259222183833575182726931236, −7.938209821736020500333671938291, −7.928715437054607046535333197668, −7.47639678854518691721782362895, −7.06720480429360381235844814450, −6.36468669123078707658828234681, −6.22162809914021680799341923990, −5.62536092260477728866504768471, −5.10655432322000120827361688973, −4.61732307188731810326957189766, −4.34033704533174739921121340134, −3.59357135158508647787518783510, −2.76494285350949537120585963927, −2.73021078120863625855221750502, −1.94348077187133837272790771925, −1.02332194949292890214877486170, −0.47797758277069428743367883195, 0.47797758277069428743367883195, 1.02332194949292890214877486170, 1.94348077187133837272790771925, 2.73021078120863625855221750502, 2.76494285350949537120585963927, 3.59357135158508647787518783510, 4.34033704533174739921121340134, 4.61732307188731810326957189766, 5.10655432322000120827361688973, 5.62536092260477728866504768471, 6.22162809914021680799341923990, 6.36468669123078707658828234681, 7.06720480429360381235844814450, 7.47639678854518691721782362895, 7.928715437054607046535333197668, 7.938209821736020500333671938291, 8.688259222183833575182726931236, 8.845823712072260614020394633224, 9.353210459151241721163483888227, 9.383427021460102407093609518027

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