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Properties

Label 4-332960-1.1-c1e2-0-0
Degree $4$
Conductor $332960$
Sign $-1$
Analytic cond. $21.2298$
Root an. cond. $2.14652$
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  − 2-s + 4-s − 2·5-s − 8-s − 9-s + 2·10-s + 13-s + 16-s − 3·17-s + 18-s − 2·20-s + 2·25-s − 26-s − 32-s + 3·34-s − 36-s − 3·37-s + 2·40-s + 12·41-s + 2·45-s − 10·49-s − 2·50-s + 52-s + 10·53-s − 6·61-s + 64-s − 2·65-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s − 1/3·9-s + 0.632·10-s + 0.277·13-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.447·20-s + 2/5·25-s − 0.196·26-s − 0.176·32-s + 0.514·34-s − 1/6·36-s − 0.493·37-s + 0.316·40-s + 1.87·41-s + 0.298·45-s − 1.42·49-s − 0.282·50-s + 0.138·52-s + 1.37·53-s − 0.768·61-s + 1/8·64-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 332960 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332960 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(332960\)    =    \(2^{5} \cdot 5 \cdot 2081\)
Sign: $-1$
Analytic conductor: \(21.2298\)
Root analytic conductor: \(2.14652\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 332960,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 3 T + p T^{2} ) \)
2081$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 31 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.11.a_ai
13$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) 2.13.ab_ba
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.17.d_y
19$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.19.a_m
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.23.a_bj
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2^2$ \( 1 - 27 T^{2} + p^{2} T^{4} \) 2.31.a_abb
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.d_bi
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.am_de
43$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.43.a_cm
47$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.47.a_abu
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.53.ak_es
59$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.59.a_ax
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.g_ec
67$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.67.a_as
71$C_2^2$ \( 1 - 124 T^{2} + p^{2} T^{4} \) 2.71.a_aeu
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.ag_ec
79$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.79.a_bm
83$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.83.a_acn
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.89.b_fg
97$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.d_gk
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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

−8.663200395145345360316246690848, −7.933364580917748606966162293179, −7.85513948703511211792961330422, −7.18261399940658997788345845481, −6.80949583979181537911039168850, −6.27618315428372144700133690228, −5.77824191749491969796772092790, −5.19129796203751530507088472739, −4.50322620585151522870908808738, −4.04663868468397143607152519881, −3.42603768023620072648995424012, −2.78391021983003531751606755209, −2.12573658758820965647274234442, −1.11696293520194761764481779157, 0, 1.11696293520194761764481779157, 2.12573658758820965647274234442, 2.78391021983003531751606755209, 3.42603768023620072648995424012, 4.04663868468397143607152519881, 4.50322620585151522870908808738, 5.19129796203751530507088472739, 5.77824191749491969796772092790, 6.27618315428372144700133690228, 6.80949583979181537911039168850, 7.18261399940658997788345845481, 7.85513948703511211792961330422, 7.933364580917748606966162293179, 8.663200395145345360316246690848

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