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Properties

Label 4-46208-1.1-c1e2-0-4
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  − 5·9-s + 2·13-s − 10·17-s − 10·25-s − 6·29-s + 4·37-s − 16·41-s − 5·49-s + 18·53-s + 28·61-s + 18·73-s + 16·81-s − 24·89-s + 28·97-s − 28·101-s + 14·109-s − 36·113-s − 10·117-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 50·153-s + 157-s + ⋯
L(s)  = 1  − 5/3·9-s + 0.554·13-s − 2.42·17-s − 2·25-s − 1.11·29-s + 0.657·37-s − 2.49·41-s − 5/7·49-s + 2.47·53-s + 3.58·61-s + 2.10·73-s + 16/9·81-s − 2.54·89-s + 2.84·97-s − 2.78·101-s + 1.34·109-s − 3.38·113-s − 0.924·117-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 4.04·153-s + 0.0798·157-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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.17.k_ch
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.41.q_fq
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.59.a_en
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.61.abc_mg
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.a_abj
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.a_bq
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.73.as_it
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.83.a_co
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−9.815161388517683585055924650409, −9.389846141086456194104312076880, −8.649214735442593578398627320691, −8.517458100677749789097011376696, −8.058703950373762851562620749027, −7.18297506166153508271072468025, −6.61570751747134755245908356006, −6.21150502900434643631796858377, −5.41676012615338774687665516924, −5.18915175026014233079805145903, −3.93346432196685798564362812764, −3.80299080464445309990426195305, −2.56005012019129152265351485884, −2.06113805202725322767384962385, 0, 2.06113805202725322767384962385, 2.56005012019129152265351485884, 3.80299080464445309990426195305, 3.93346432196685798564362812764, 5.18915175026014233079805145903, 5.41676012615338774687665516924, 6.21150502900434643631796858377, 6.61570751747134755245908356006, 7.18297506166153508271072468025, 8.058703950373762851562620749027, 8.517458100677749789097011376696, 8.649214735442593578398627320691, 9.389846141086456194104312076880, 9.815161388517683585055924650409

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