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Properties

Label 4-85536-1.1-c1e2-0-8
Degree $4$
Conductor $85536$
Sign $-1$
Analytic cond. $5.45384$
Root an. cond. $1.52818$
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 + 8-s − 6·11-s − 11·13-s + 16-s − 6·22-s − 6·23-s + 3·25-s − 11·26-s + 32-s + 37-s − 6·44-s − 6·46-s − 5·47-s + 6·49-s + 3·50-s − 11·52-s − 3·59-s + 11·61-s + 64-s − 15·71-s − 5·73-s + 74-s − 10·83-s − 6·88-s − 6·92-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s − 3.05·13-s + 1/4·16-s − 1.27·22-s − 1.25·23-s + 3/5·25-s − 2.15·26-s + 0.176·32-s + 0.164·37-s − 0.904·44-s − 0.884·46-s − 0.729·47-s + 6/7·49-s + 0.424·50-s − 1.52·52-s − 0.390·59-s + 1.40·61-s + 1/8·64-s − 1.78·71-s − 0.585·73-s + 0.116·74-s − 1.09·83-s − 0.639·88-s − 0.625·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(85536\)    =    \(2^{5} \cdot 3^{5} \cdot 11\)
Sign: $-1$
Analytic conductor: \(5.45384\)
Root analytic conductor: \(1.52818\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 85536,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3 \( 1 \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 5 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.5.a_ad
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
13$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.l_ce
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.17.a_au
19$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \) 2.19.a_bb
23$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.23.g_bz
29$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.29.a_z
31$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \) 2.31.a_bk
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.ab_ck
41$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.41.a_abc
43$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.43.a_t
47$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.f_cs
53$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.53.a_abi
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.d_eo
61$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + p T^{2} ) \) 2.61.al_es
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.p_fm
73$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.f_u
79$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.79.a_cq
83$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.83.k_w
89$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \) 2.89.a_ds
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 16 T + p T^{2} ) \) 2.97.abi_so
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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

−9.739693672523050389409171493176, −8.948845034257412493544188329211, −8.275473806967943463795186412142, −7.71631479687046193607097414839, −7.42535968172751911317996018843, −7.06021940370550923367716412230, −6.25957672423399325415475212681, −5.58754757540114519899709041270, −5.12303808685008881439439388970, −4.76659099561030893421825165230, −4.18592265784892316981748693244, −3.11272914247628064232761269405, −2.55559813863456797393839233628, −2.12275984284545528487513315757, 0, 2.12275984284545528487513315757, 2.55559813863456797393839233628, 3.11272914247628064232761269405, 4.18592265784892316981748693244, 4.76659099561030893421825165230, 5.12303808685008881439439388970, 5.58754757540114519899709041270, 6.25957672423399325415475212681, 7.06021940370550923367716412230, 7.42535968172751911317996018843, 7.71631479687046193607097414839, 8.275473806967943463795186412142, 8.948845034257412493544188329211, 9.739693672523050389409171493176

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