Properties

Label 4-3312e2-1.1-c1e2-0-8
Degree $4$
Conductor $10969344$
Sign $1$
Analytic cond. $699.414$
Root an. cond. $5.14261$
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  − 4·5-s + 2·11-s + 5·13-s − 2·17-s − 2·19-s − 2·23-s + 2·25-s − 3·29-s + 9·31-s − 41-s + 16·43-s + 11·47-s − 14·49-s − 4·53-s − 8·55-s + 4·59-s + 8·61-s − 20·65-s + 2·67-s + 23·71-s − 17·73-s + 2·79-s + 12·83-s + 8·85-s + 2·89-s + 8·95-s − 2·97-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.603·11-s + 1.38·13-s − 0.485·17-s − 0.458·19-s − 0.417·23-s + 2/5·25-s − 0.557·29-s + 1.61·31-s − 0.156·41-s + 2.43·43-s + 1.60·47-s − 2·49-s − 0.549·53-s − 1.07·55-s + 0.520·59-s + 1.02·61-s − 2.48·65-s + 0.244·67-s + 2.72·71-s − 1.98·73-s + 0.225·79-s + 1.31·83-s + 0.867·85-s + 0.211·89-s + 0.820·95-s − 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10969344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10969344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10969344\)    =    \(2^{8} \cdot 3^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(699.414\)
Root analytic conductor: \(5.14261\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10969344,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.976194467\)
\(L(\frac12)\) \(\approx\) \(1.976194467\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_g
13$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.13.af_bc
17$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_s
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_w
29$D_{4}$ \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_ce
31$D_{4}$ \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.31.aj_da
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.37.a_g
41$D_{4}$ \( 1 + T - 24 T^{2} + p T^{3} + p^{2} T^{4} \) 2.41.b_ay
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$D_{4}$ \( 1 - 11 T + 86 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.47.al_di
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_cc
61$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_cs
67$D_{4}$ \( 1 - 2 T + 118 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_eo
71$D_{4}$ \( 1 - 23 T + 270 T^{2} - 23 p T^{3} + p^{2} T^{4} \) 2.71.ax_kk
73$D_{4}$ \( 1 + 17 T + 180 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.73.r_gy
79$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_fm
83$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_fe
89$D_{4}$ \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.89.ac_ba
97$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_bq
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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

−8.535185484525609968501016157686, −8.484652865828331613722270960161, −8.116282381550409061528746723874, −7.73580135228354232855793029670, −7.44229362665304822290993574684, −7.03716308732209282300110852364, −6.46915736737043775846941667305, −6.36900333951428284812280623387, −5.75026148095619163971865868535, −5.62550360975466260795004550715, −4.63465771887283410881463454096, −4.56005859389813946471320411952, −4.05740073099969839511991483442, −3.85227798450954084882438326413, −3.38799005375643845830239346399, −3.03936951224004166450175734061, −2.16336592544872534607461564469, −1.89323272175238904203744384988, −0.826726240466976182437078614861, −0.61040374913990718689447071714, 0.61040374913990718689447071714, 0.826726240466976182437078614861, 1.89323272175238904203744384988, 2.16336592544872534607461564469, 3.03936951224004166450175734061, 3.38799005375643845830239346399, 3.85227798450954084882438326413, 4.05740073099969839511991483442, 4.56005859389813946471320411952, 4.63465771887283410881463454096, 5.62550360975466260795004550715, 5.75026148095619163971865868535, 6.36900333951428284812280623387, 6.46915736737043775846941667305, 7.03716308732209282300110852364, 7.44229362665304822290993574684, 7.73580135228354232855793029670, 8.116282381550409061528746723874, 8.484652865828331613722270960161, 8.535185484525609968501016157686

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