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Properties

Label 4-585e2-1.1-c1e2-0-13
Degree $4$
Conductor $342225$
Sign $1$
Analytic cond. $21.8205$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·5-s + 4·7-s + 6·11-s + 2·13-s − 3·16-s − 2·19-s − 2·20-s − 6·23-s + 3·25-s − 4·28-s + 12·29-s + 10·31-s + 8·35-s − 8·37-s + 10·43-s − 6·44-s − 12·47-s − 2·49-s − 2·52-s + 12·55-s + 6·59-s + 4·61-s + 7·64-s + 4·65-s − 8·67-s − 6·71-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 1.51·7-s + 1.80·11-s + 0.554·13-s − 3/4·16-s − 0.458·19-s − 0.447·20-s − 1.25·23-s + 3/5·25-s − 0.755·28-s + 2.22·29-s + 1.79·31-s + 1.35·35-s − 1.31·37-s + 1.52·43-s − 0.904·44-s − 1.75·47-s − 2/7·49-s − 0.277·52-s + 1.61·55-s + 0.781·59-s + 0.512·61-s + 7/8·64-s + 0.496·65-s − 0.977·67-s − 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(342225\)    =    \(3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(21.8205\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 342225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.723926569\)
\(L(\frac12)\) \(\approx\) \(2.723926569\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_m
23$D_{4}$ \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_ca
29$D_{4}$ \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_de
31$D_{4}$ \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_ci
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_dg
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.53.a_ac
59$D_{4}$ \( 1 - 6 T - 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_au
61$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_s
67$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_bq
71$D_{4}$ \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_fs
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_cc
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_gk
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−10.94472824657291446880371885590, −10.45990677787730811948855956358, −9.877348407940449699969248205996, −9.789134272976399542526438593140, −8.904475350883832398689412944219, −8.884182644576086580948049551444, −8.307523374498070068525019628212, −8.113862234208344136020896150558, −7.38013590555605613070685443106, −6.56456973101159506347002258038, −6.42499351060238341075416544134, −6.15119478756246662288907107930, −5.16455123203548927403929524785, −4.92456319302782996777442433173, −4.25102037691181755072506709673, −4.11891521514934202622270260694, −3.13978927529927978171053329719, −2.30257434020388696837873314714, −1.63717590887318719464495311003, −1.09175533008117792775729773952, 1.09175533008117792775729773952, 1.63717590887318719464495311003, 2.30257434020388696837873314714, 3.13978927529927978171053329719, 4.11891521514934202622270260694, 4.25102037691181755072506709673, 4.92456319302782996777442433173, 5.16455123203548927403929524785, 6.15119478756246662288907107930, 6.42499351060238341075416544134, 6.56456973101159506347002258038, 7.38013590555605613070685443106, 8.113862234208344136020896150558, 8.307523374498070068525019628212, 8.884182644576086580948049551444, 8.904475350883832398689412944219, 9.789134272976399542526438593140, 9.877348407940449699969248205996, 10.45990677787730811948855956358, 10.94472824657291446880371885590

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