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Properties

Label 4-1404e2-1.1-c1e2-0-22
Degree $4$
Conductor $1971216$
Sign $1$
Analytic cond. $125.686$
Root an. cond. $3.34828$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 2·7-s − 3·11-s − 2·13-s − 3·17-s + 3·19-s − 9·23-s − 6·29-s − 4·31-s − 6·35-s − 6·41-s + 5·43-s − 9·47-s − 11·49-s − 3·53-s + 9·55-s − 12·59-s − 4·61-s + 6·65-s − 7·67-s − 6·71-s + 73-s − 6·77-s + 19·79-s − 18·83-s + 9·85-s + 6·89-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.755·7-s − 0.904·11-s − 0.554·13-s − 0.727·17-s + 0.688·19-s − 1.87·23-s − 1.11·29-s − 0.718·31-s − 1.01·35-s − 0.937·41-s + 0.762·43-s − 1.31·47-s − 1.57·49-s − 0.412·53-s + 1.21·55-s − 1.56·59-s − 0.512·61-s + 0.744·65-s − 0.855·67-s − 0.712·71-s + 0.117·73-s − 0.683·77-s + 2.13·79-s − 1.97·83-s + 0.976·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1971216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1971216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1971216\)    =    \(2^{4} \cdot 3^{6} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(125.686\)
Root analytic conductor: \(3.34828\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1971216,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_j
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.7.ac_p
11$D_{4}$ \( 1 + 3 T + 21 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_v
17$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_h
19$D_{4}$ \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.19.ad_l
23$D_{4}$ \( 1 + 9 T + 63 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.23.j_cl
29$D_{4}$ \( 1 + 6 T + 15 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_p
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2^2$ \( 1 - 43 T^{2} + p^{2} T^{4} \) 2.37.a_abr
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.43.af_cl
47$D_{4}$ \( 1 + 9 T + 111 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.47.j_eh
53$D_{4}$ \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_db
59$D_{4}$ \( 1 + 12 T + 141 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fl
61$D_{4}$ \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_j
67$D_{4}$ \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_en
71$D_{4}$ \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_fi
73$D_{4}$ \( 1 - T + 117 T^{2} - p T^{3} + p^{2} T^{4} \) 2.73.ab_en
79$D_{4}$ \( 1 - 19 T + 219 T^{2} - 19 p T^{3} + p^{2} T^{4} \) 2.79.at_il
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.83.s_jn
89$D_{4}$ \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_cs
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−9.170990449002618408949115550110, −9.127668641372333868128385474050, −8.242049139397797810929312232866, −8.068271856751349530769317281343, −7.72076400875542773143032118716, −7.68557484249060732332764896365, −7.00941699965204337380847414721, −6.61747023784168790747845727679, −5.91762257961329957573345362985, −5.67784640855690457163003189071, −4.92710433281573828647292322949, −4.80692711665144772153765495655, −4.20548974431761864214088935941, −3.80089156441554848134430383586, −3.29848760807981930369072729166, −2.74347357420448619178518541704, −1.91990089533593992240366673691, −1.62385647385633735947417246016, 0, 0, 1.62385647385633735947417246016, 1.91990089533593992240366673691, 2.74347357420448619178518541704, 3.29848760807981930369072729166, 3.80089156441554848134430383586, 4.20548974431761864214088935941, 4.80692711665144772153765495655, 4.92710433281573828647292322949, 5.67784640855690457163003189071, 5.91762257961329957573345362985, 6.61747023784168790747845727679, 7.00941699965204337380847414721, 7.68557484249060732332764896365, 7.72076400875542773143032118716, 8.068271856751349530769317281343, 8.242049139397797810929312232866, 9.127668641372333868128385474050, 9.170990449002618408949115550110

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