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Properties

Label 4-3276e2-1.1-c1e2-0-40
Degree $4$
Conductor $10732176$
Sign $1$
Analytic cond. $684.292$
Root an. cond. $5.11458$
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  + 2·5-s − 2·7-s − 8·11-s − 2·13-s + 6·19-s + 2·23-s − 25-s + 2·29-s + 2·31-s − 4·35-s − 12·37-s − 12·41-s + 2·43-s − 2·47-s + 3·49-s + 6·53-s − 16·55-s − 28·59-s − 4·61-s − 4·65-s − 4·67-s − 20·71-s − 2·73-s + 16·77-s − 14·79-s − 6·83-s + 18·89-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s − 2.41·11-s − 0.554·13-s + 1.37·19-s + 0.417·23-s − 1/5·25-s + 0.371·29-s + 0.359·31-s − 0.676·35-s − 1.97·37-s − 1.87·41-s + 0.304·43-s − 0.291·47-s + 3/7·49-s + 0.824·53-s − 2.15·55-s − 3.64·59-s − 0.512·61-s − 0.496·65-s − 0.488·67-s − 2.37·71-s − 0.234·73-s + 1.82·77-s − 1.57·79-s − 0.658·83-s + 1.90·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10732176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10732176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10732176\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(684.292\)
Root analytic conductor: \(5.11458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 10732176,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_f
11$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.11.i_bg
17$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.17.a_bc
19$D_{4}$ \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_bp
23$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_x
29$D_{4}$ \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_bj
31$D_{4}$ \( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_cf
37$D_{4}$ \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.37.m_ea
41$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dq
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.43.ac_dj
47$D_{4}$ \( 1 + 2 T + 89 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_dl
53$D_{4}$ \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_dn
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.59.bc_mc
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$D_{4}$ \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_ek
71$D_{4}$ \( 1 + 20 T + 236 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.71.u_jc
73$D_{4}$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_ad
79$D_{4}$ \( 1 + 14 T + 183 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_hb
83$D_{4}$ \( 1 + 6 T + 121 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_er
89$D_{4}$ \( 1 - 18 T + 253 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_jt
97$D_{4}$ \( 1 + 26 T + 357 T^{2} + 26 p T^{3} + p^{2} T^{4} \) 2.97.ba_nt
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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

−8.337738487738014681335711182358, −8.184088296645891185267626658223, −7.53692710886065655935655547019, −7.38894253982369871605140801410, −7.10568527129737175282689633748, −6.58720921063941438349719820489, −6.03362920310515635499047521570, −5.84668565958769368590259303236, −5.29820245079504364250535064962, −5.19675108668668633859110348244, −4.73292263415156432634225485336, −4.34237360373598366363444728277, −3.41608628272698231614099771862, −3.19869750438025785505847286358, −2.78696056389891760719970121886, −2.50706249393881370898857710437, −1.71439641404488275988091424081, −1.36689829670193922323928913058, 0, 0, 1.36689829670193922323928913058, 1.71439641404488275988091424081, 2.50706249393881370898857710437, 2.78696056389891760719970121886, 3.19869750438025785505847286358, 3.41608628272698231614099771862, 4.34237360373598366363444728277, 4.73292263415156432634225485336, 5.19675108668668633859110348244, 5.29820245079504364250535064962, 5.84668565958769368590259303236, 6.03362920310515635499047521570, 6.58720921063941438349719820489, 7.10568527129737175282689633748, 7.38894253982369871605140801410, 7.53692710886065655935655547019, 8.184088296645891185267626658223, 8.337738487738014681335711182358

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