Properties

Label 4-195e2-1.1-c1e2-0-13
Degree $4$
Conductor $38025$
Sign $1$
Analytic cond. $2.42450$
Root an. cond. $1.24783$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·4-s − 8·7-s + 9-s + 6·12-s − 2·13-s + 5·16-s − 12·19-s + 16·21-s + 25-s + 4·27-s + 24·28-s − 20·31-s − 3·36-s − 4·37-s + 4·39-s + 20·43-s − 10·48-s + 34·49-s + 6·52-s + 24·57-s + 4·61-s − 8·63-s − 3·64-s − 8·67-s − 12·73-s − 2·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s − 3.02·7-s + 1/3·9-s + 1.73·12-s − 0.554·13-s + 5/4·16-s − 2.75·19-s + 3.49·21-s + 1/5·25-s + 0.769·27-s + 4.53·28-s − 3.59·31-s − 1/2·36-s − 0.657·37-s + 0.640·39-s + 3.04·43-s − 1.44·48-s + 34/7·49-s + 0.832·52-s + 3.17·57-s + 0.512·61-s − 1.00·63-s − 3/8·64-s − 0.977·67-s − 1.40·73-s − 0.230·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(38025\)    =    \(3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2.42450\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 38025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−9.844976583062747102624785032359, −9.238371497422229278947323218386, −8.903866275348653436076253219750, −8.679150483937958571354498819427, −7.36945570129601409328488904448, −7.04231773996815600274218157258, −6.35831289645710659676663930685, −5.80677556175820100036715333304, −5.66957165960173077695586779200, −4.59731852050444182415870036677, −4.03550878858777485512286959394, −3.51994456797006266949922249859, −2.53409816778618263405222233425, 0, 0, 2.53409816778618263405222233425, 3.51994456797006266949922249859, 4.03550878858777485512286959394, 4.59731852050444182415870036677, 5.66957165960173077695586779200, 5.80677556175820100036715333304, 6.35831289645710659676663930685, 7.04231773996815600274218157258, 7.36945570129601409328488904448, 8.679150483937958571354498819427, 8.903866275348653436076253219750, 9.238371497422229278947323218386, 9.844976583062747102624785032359

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