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Properties

Label 4-20032-1.1-c1e2-0-0
Degree $4$
Conductor $20032$
Sign $-1$
Analytic cond. $1.27725$
Root an. cond. $1.06308$
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 − 2·7-s + 3·8-s − 4·9-s + 2·14-s − 16-s + 3·17-s + 4·18-s − 8·25-s + 2·28-s − 5·31-s − 5·32-s − 3·34-s + 4·36-s − 19·41-s − 2·47-s − 7·49-s + 8·50-s − 6·56-s + 5·62-s + 8·63-s + 7·64-s − 3·68-s − 6·71-s − 12·72-s + 17·79-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s − 4/3·9-s + 0.534·14-s − 1/4·16-s + 0.727·17-s + 0.942·18-s − 8/5·25-s + 0.377·28-s − 0.898·31-s − 0.883·32-s − 0.514·34-s + 2/3·36-s − 2.96·41-s − 0.291·47-s − 49-s + 1.13·50-s − 0.801·56-s + 0.635·62-s + 1.00·63-s + 7/8·64-s − 0.363·68-s − 0.712·71-s − 1.41·72-s + 1.91·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20032\)    =    \(2^{6} \cdot 313\)
Sign: $-1$
Analytic conductor: \(1.27725\)
Root analytic conductor: \(1.06308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 20032,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
313$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 22 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
7$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.c_l
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ad_bi
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.23.a_v
29$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.29.a_abp
31$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.f_bm
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.41.t_go
43$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.43.a_h
47$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.c_dn
53$C_2^2$ \( 1 - 60 T^{2} + p^{2} T^{4} \) 2.53.a_aci
59$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.59.a_bj
61$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.61.a_ei
67$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.67.a_ar
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.g_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.79.ar_gs
83$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.83.a_ack
89$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.t_jo
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.97.as_jq
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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

−10.35232913289083869072810666947, −9.961766708988868119662710952421, −9.596122649841018551132494920764, −8.931518388640699156060907930920, −8.515622132681417924371635300806, −7.998981405432442683631264656800, −7.46545999334846420694712129804, −6.69427157872256477469464923191, −5.99132808556502771544033579895, −5.42546455578380926442486730023, −4.79026160286546843766610641541, −3.65810041371993239605100688559, −3.25265154413136267114715079332, −1.88789585197718814951113118824, 0, 1.88789585197718814951113118824, 3.25265154413136267114715079332, 3.65810041371993239605100688559, 4.79026160286546843766610641541, 5.42546455578380926442486730023, 5.99132808556502771544033579895, 6.69427157872256477469464923191, 7.46545999334846420694712129804, 7.998981405432442683631264656800, 8.515622132681417924371635300806, 8.931518388640699156060907930920, 9.596122649841018551132494920764, 9.961766708988868119662710952421, 10.35232913289083869072810666947

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