Properties

Label 4-3840e2-1.1-c1e2-0-37
Degree $4$
Conductor $14745600$
Sign $1$
Analytic cond. $940.192$
Root an. cond. $5.53737$
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  − 2·3-s + 4·5-s + 3·9-s − 8·15-s + 11·25-s − 4·27-s + 8·31-s + 16·37-s + 20·41-s + 8·43-s + 12·45-s − 2·49-s − 24·53-s − 8·67-s − 22·75-s + 24·79-s + 5·81-s − 8·83-s − 20·89-s − 16·93-s − 24·107-s − 32·111-s + 6·121-s − 40·123-s + 24·125-s + 127-s − 16·129-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 9-s − 2.06·15-s + 11/5·25-s − 0.769·27-s + 1.43·31-s + 2.63·37-s + 3.12·41-s + 1.21·43-s + 1.78·45-s − 2/7·49-s − 3.29·53-s − 0.977·67-s − 2.54·75-s + 2.70·79-s + 5/9·81-s − 0.878·83-s − 2.11·89-s − 1.65·93-s − 2.32·107-s − 3.03·111-s + 6/11·121-s − 3.60·123-s + 2.14·125-s + 0.0887·127-s − 1.40·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(940.192\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.148072913\)
\(L(\frac12)\) \(\approx\) \(3.148072913\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.849374337804467343214093047852, −8.100339686624105484240114584092, −7.929210961192851874185850441225, −7.63731906834285617515929822828, −7.01765088310070113605281841117, −6.65894874970356530470862956954, −6.21814891903146690481519422026, −6.15116493275716350907595604770, −5.70920482603491429739545368637, −5.58998272192695646335070045087, −4.88304276254778416050683265833, −4.48843092112184210901224627671, −4.46442796500013484512843665417, −3.78725566099458640682077395931, −2.84455365631458794571511144179, −2.76640496182648206946222259235, −2.27108252912912709956622885863, −1.54984243871760249027867512690, −1.13379911430019667683309605047, −0.63134009277517368460804155646, 0.63134009277517368460804155646, 1.13379911430019667683309605047, 1.54984243871760249027867512690, 2.27108252912912709956622885863, 2.76640496182648206946222259235, 2.84455365631458794571511144179, 3.78725566099458640682077395931, 4.46442796500013484512843665417, 4.48843092112184210901224627671, 4.88304276254778416050683265833, 5.58998272192695646335070045087, 5.70920482603491429739545368637, 6.15116493275716350907595604770, 6.21814891903146690481519422026, 6.65894874970356530470862956954, 7.01765088310070113605281841117, 7.63731906834285617515929822828, 7.929210961192851874185850441225, 8.100339686624105484240114584092, 8.849374337804467343214093047852

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