Properties

Label 4-2352e2-1.1-c1e2-0-49
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $352.718$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s − 4·19-s + 7·25-s + 4·27-s + 18·29-s + 10·31-s + 20·37-s + 24·47-s − 18·53-s − 8·57-s − 18·59-s + 14·75-s + 5·81-s + 6·83-s + 36·87-s + 20·93-s − 8·103-s − 8·109-s + 40·111-s + 12·113-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 48·141-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 0.917·19-s + 7/5·25-s + 0.769·27-s + 3.34·29-s + 1.79·31-s + 3.28·37-s + 3.50·47-s − 2.47·53-s − 1.05·57-s − 2.34·59-s + 1.61·75-s + 5/9·81-s + 0.658·83-s + 3.85·87-s + 2.07·93-s − 0.788·103-s − 0.766·109-s + 3.79·111-s + 1.12·113-s + 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.04·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(352.718\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.327382774\)
\(L(\frac12)\) \(\approx\) \(5.327382774\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 166 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−9.001459039225397070308812829286, −8.852419166544719296225306386758, −8.282087060702713190370508050758, −8.212846295492343883968322783125, −7.59730443176789810781247623848, −7.58494722152220269601368204874, −6.65030428684729264818970089276, −6.62233196555210255726968134397, −6.17343778876993231085065776638, −5.86702366733265572321818160965, −4.92270996042427055340456072908, −4.66511463529891051653281781832, −4.31648877412164219233067957796, −4.14301318316900065635135627308, −3.09280100643736370452388762144, −2.92619016772606804013986592059, −2.64127991746733602747381643297, −2.10302338141850890160574972475, −1.01962032116672229306102701652, −0.962526604874529470469330704873, 0.962526604874529470469330704873, 1.01962032116672229306102701652, 2.10302338141850890160574972475, 2.64127991746733602747381643297, 2.92619016772606804013986592059, 3.09280100643736370452388762144, 4.14301318316900065635135627308, 4.31648877412164219233067957796, 4.66511463529891051653281781832, 4.92270996042427055340456072908, 5.86702366733265572321818160965, 6.17343778876993231085065776638, 6.62233196555210255726968134397, 6.65030428684729264818970089276, 7.58494722152220269601368204874, 7.59730443176789810781247623848, 8.212846295492343883968322783125, 8.282087060702713190370508050758, 8.852419166544719296225306386758, 9.001459039225397070308812829286

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