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Properties

Label 4-46208-1.1-c1e2-0-5
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-s + 2·3-s + 4-s − 2·6-s − 8-s − 3·9-s − 12·11-s + 2·12-s + 16-s + 6·17-s + 3·18-s + 2·19-s + 12·22-s − 2·24-s − 10·25-s − 14·27-s − 32-s − 24·33-s − 6·34-s − 3·36-s − 2·38-s + 16·43-s − 12·44-s + 2·48-s − 13·49-s + 10·50-s + 12·51-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s − 9-s − 3.61·11-s + 0.577·12-s + 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.458·19-s + 2.55·22-s − 0.408·24-s − 2·25-s − 2.69·27-s − 0.176·32-s − 4.17·33-s − 1.02·34-s − 1/2·36-s − 0.324·38-s + 2.43·43-s − 1.80·44-s + 0.288·48-s − 1.85·49-s + 1.41·50-s + 1.68·51-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$C_1$ \( 1 + T \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.3.ac_h
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.a_dt
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.59.as_hr
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.67.ak_gd
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−9.937093458417137271218639490647, −9.488910467190776994037417053794, −8.684413720393242419277779815238, −8.165660732182309128185574185775, −7.944498198632498175704416365123, −7.73591843742988285704805588989, −7.14449686564876148233582580493, −5.81027585652647529822118508915, −5.56559135650201084358170066267, −5.32600053322851104543262796765, −3.98931312476793193392332292270, −2.99432139105097413830284496966, −2.80468657266648836263703727877, −2.16004950277651645615210004057, 0, 2.16004950277651645615210004057, 2.80468657266648836263703727877, 2.99432139105097413830284496966, 3.98931312476793193392332292270, 5.32600053322851104543262796765, 5.56559135650201084358170066267, 5.81027585652647529822118508915, 7.14449686564876148233582580493, 7.73591843742988285704805588989, 7.944498198632498175704416365123, 8.165660732182309128185574185775, 8.684413720393242419277779815238, 9.488910467190776994037417053794, 9.937093458417137271218639490647

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