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Properties

Label 4-49664-1.1-c1e2-0-5
Degree $4$
Conductor $49664$
Sign $-1$
Analytic cond. $3.16661$
Root an. cond. $1.33397$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·9-s − 8·13-s + 3·17-s − 9·25-s + 7·29-s − 6·37-s − 9·41-s − 12·49-s + 17·53-s − 5·61-s − 19·73-s + 7·81-s + 2·89-s − 9·97-s − 6·101-s + 2·109-s − 6·113-s + 32·117-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + ⋯
L(s)  = 1  − 4/3·9-s − 2.21·13-s + 0.727·17-s − 9/5·25-s + 1.29·29-s − 0.986·37-s − 1.40·41-s − 1.71·49-s + 2.33·53-s − 0.640·61-s − 2.22·73-s + 7/9·81-s + 0.211·89-s − 0.913·97-s − 0.597·101-s + 0.191·109-s − 0.564·113-s + 2.95·117-s + 1.81·121-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 − 0.970·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(49664\)    =    \(2^{9} \cdot 97\)
Sign: $-1$
Analytic conductor: \(3.16661\)
Root analytic conductor: \(1.33397\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 49664,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
97$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 10 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.7.a_m
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.11.a_au
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ad_bi
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.23.a_q
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.29.ah_cm
31$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.31.a_av
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.g_t
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.j_de
43$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.43.a_abe
47$C_2^2$ \( 1 - 83 T^{2} + p^{2} T^{4} \) 2.47.a_adf
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.53.ar_gw
59$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.59.a_n
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.61.f_es
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.67.a_bi
71$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \) 2.71.a_ea
73$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.t_ii
79$C_2^2$ \( 1 - 69 T^{2} + p^{2} T^{4} \) 2.79.a_acr
83$C_2^2$ \( 1 - 139 T^{2} + p^{2} T^{4} \) 2.83.a_afj
89$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.ac_db
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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

−9.841192558366757056084420708126, −9.530602295912853524039239928155, −8.656638269600292156936983986043, −8.401730273738019127211198593677, −7.76976761684010009289798773199, −7.27208436733779362677998890965, −6.76032811962280527874055419018, −5.95546271658245618502359203416, −5.48461663042888163893330989684, −4.98594499313067182781186396742, −4.31346322633020253857150750755, −3.31868673028657931700004811950, −2.77451040342111329664053932159, −1.94543910615696519736755824230, 0, 1.94543910615696519736755824230, 2.77451040342111329664053932159, 3.31868673028657931700004811950, 4.31346322633020253857150750755, 4.98594499313067182781186396742, 5.48461663042888163893330989684, 5.95546271658245618502359203416, 6.76032811962280527874055419018, 7.27208436733779362677998890965, 7.76976761684010009289798773199, 8.401730273738019127211198593677, 8.656638269600292156936983986043, 9.530602295912853524039239928155, 9.841192558366757056084420708126

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