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Properties

Label 4-912e2-1.1-c1e2-0-77
Degree $4$
Conductor $831744$
Sign $1$
Analytic cond. $53.0327$
Root an. cond. $2.69858$
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 + 8·7-s + 9-s − 2·19-s + 16·21-s + 8·25-s − 4·27-s + 12·29-s − 4·43-s + 34·49-s − 12·53-s − 4·57-s + 24·59-s + 4·61-s + 8·63-s − 24·71-s − 8·73-s + 16·75-s − 11·81-s + 24·87-s − 12·89-s − 24·107-s + 24·113-s − 10·121-s + 127-s − 8·129-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s + 3.02·7-s + 1/3·9-s − 0.458·19-s + 3.49·21-s + 8/5·25-s − 0.769·27-s + 2.22·29-s − 0.609·43-s + 34/7·49-s − 1.64·53-s − 0.529·57-s + 3.12·59-s + 0.512·61-s + 1.00·63-s − 2.84·71-s − 0.936·73-s + 1.84·75-s − 1.22·81-s + 2.57·87-s − 1.27·89-s − 2.32·107-s + 2.25·113-s − 0.909·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(831744\)    =    \(2^{8} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(53.0327\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 831744,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.939707428\)
\(L(\frac12)\) \(\approx\) \(4.939707428\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
19$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.13.a_ai
17$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.17.a_aba
23$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.23.a_abs
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.31.a_abs
37$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \) 2.37.a_ace
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.43.e_dm
47$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.47.a_ado
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.67.a_afe
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \) 2.79.a_afk
83$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.83.a_agc
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \) 2.97.a_aes
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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

−10.28509335105820461948263180658, −9.978671065961800473017045413671, −9.300069343039410276111258811739, −8.668407020429631791837187438692, −8.485436469931972758114408459179, −8.444961022619276800782710193718, −7.975050154688608856900829809602, −7.51681246314878098211096912736, −7.05096874502569349896050384171, −6.64270932641587577872259995017, −5.84669845906003147891072410212, −5.36109871006620078070576367038, −4.82959454165294789558560807376, −4.57640624267292969372136466235, −4.19956131266350948543110261800, −3.38603598172315336102113580230, −2.70008758134688100532996270514, −2.34271117101274267172122724383, −1.55648271257340826115003204993, −1.17183039492744993515610633033, 1.17183039492744993515610633033, 1.55648271257340826115003204993, 2.34271117101274267172122724383, 2.70008758134688100532996270514, 3.38603598172315336102113580230, 4.19956131266350948543110261800, 4.57640624267292969372136466235, 4.82959454165294789558560807376, 5.36109871006620078070576367038, 5.84669845906003147891072410212, 6.64270932641587577872259995017, 7.05096874502569349896050384171, 7.51681246314878098211096912736, 7.975050154688608856900829809602, 8.444961022619276800782710193718, 8.485436469931972758114408459179, 8.668407020429631791837187438692, 9.300069343039410276111258811739, 9.978671065961800473017045413671, 10.28509335105820461948263180658

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