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Properties

Label 4-46208-1.1-c1e2-0-3
Degree $4$
Conductor $46208$
Sign $-1$
Analytic cond. $2.94626$
Root an. cond. $1.31014$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·9-s − 8·13-s + 10·17-s − 7·25-s + 4·29-s − 20·37-s + 12·41-s + 4·45-s − 5·49-s − 16·53-s − 10·61-s + 16·65-s − 30·73-s − 5·81-s − 20·85-s + 32·97-s − 36·101-s + 24·109-s + 4·113-s + 16·117-s − 13·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.894·5-s − 2/3·9-s − 2.21·13-s + 2.42·17-s − 7/5·25-s + 0.742·29-s − 3.28·37-s + 1.87·41-s + 0.596·45-s − 5/7·49-s − 2.19·53-s − 1.28·61-s + 1.98·65-s − 3.51·73-s − 5/9·81-s − 2.16·85-s + 3.24·97-s − 3.58·101-s + 2.29·109-s + 0.376·113-s + 1.47·117-s − 1.18·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(46208\)    =    \(2^{7} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(2.94626\)
Root analytic conductor: \(1.31014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 46208,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.17.ak_ch
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.43.a_bl
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.53.q_go
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.a_ada
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.61.k_fr
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.73.be_oh
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.97.abg_ri
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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

−9.829456006504791455536603563268, −9.575094487650744885163867781494, −8.865784190712433894154140882392, −8.148422841499315186159351254036, −7.79236899027639545070642498110, −7.41110136347227768123475878009, −6.97775492717497319205239152750, −5.89152757780490661000300260672, −5.62405161804576690819108289841, −4.85172758929064856672456822692, −4.35808362404597816678481770906, −3.25598629803065603741482539939, −3.08915058635532111035563541614, −1.81226351589066908089801551987, 0, 1.81226351589066908089801551987, 3.08915058635532111035563541614, 3.25598629803065603741482539939, 4.35808362404597816678481770906, 4.85172758929064856672456822692, 5.62405161804576690819108289841, 5.89152757780490661000300260672, 6.97775492717497319205239152750, 7.41110136347227768123475878009, 7.79236899027639545070642498110, 8.148422841499315186159351254036, 8.865784190712433894154140882392, 9.575094487650744885163867781494, 9.829456006504791455536603563268

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