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Properties

Label 4-85280-1.1-c1e2-0-9
Degree $4$
Conductor $85280$
Sign $-1$
Analytic cond. $5.43752$
Root an. cond. $1.52703$
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 − 5-s + 8-s − 2·9-s − 10-s − 3·13-s + 16-s − 16·17-s − 2·18-s − 20-s − 2·25-s − 3·26-s + 6·29-s + 32-s − 16·34-s − 2·36-s + 2·37-s − 40-s − 3·41-s + 2·45-s − 10·49-s − 2·50-s − 3·52-s − 8·53-s + 6·58-s + 22·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.832·13-s + 1/4·16-s − 3.88·17-s − 0.471·18-s − 0.223·20-s − 2/5·25-s − 0.588·26-s + 1.11·29-s + 0.176·32-s − 2.74·34-s − 1/3·36-s + 0.328·37-s − 0.158·40-s − 0.468·41-s + 0.298·45-s − 1.42·49-s − 0.282·50-s − 0.416·52-s − 1.09·53-s + 0.787·58-s + 2.81·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(85280\)    =    \(2^{5} \cdot 5 \cdot 13 \cdot 41\)
Sign: $-1$
Analytic conductor: \(5.43752\)
Root analytic conductor: \(1.52703\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 85280,\ (\ :1/2, 1/2),\ -1)\)

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$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.17.q_du
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.23.a_abm
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.ag_co
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.37.ac_cw
43$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.43.a_ae
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.47.a_bu
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.i_w
59$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.59.a_abg
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.61.aw_ji
67$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \) 2.67.a_ce
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.m_gw
79$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.79.a_bm
83$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.83.a_cs
89$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.89.ae_fq
97$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.g_fy
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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

−9.445434149175178190534837201998, −8.826171809610314934244672989252, −8.408102154869648800813162339473, −8.082837341209114977953459495489, −7.11020074319077968499505533310, −6.89588461251742922722045775042, −6.40684447171308426551470593506, −5.84469969914031163846408617398, −4.98761585783446167262346255987, −4.55517410065217358457151523105, −4.23734767472938567260362670711, −3.30314266254156899036223991684, −2.51928222123784599091347165996, −2.05221934148470250098572163466, 0, 2.05221934148470250098572163466, 2.51928222123784599091347165996, 3.30314266254156899036223991684, 4.23734767472938567260362670711, 4.55517410065217358457151523105, 4.98761585783446167262346255987, 5.84469969914031163846408617398, 6.40684447171308426551470593506, 6.89588461251742922722045775042, 7.11020074319077968499505533310, 8.082837341209114977953459495489, 8.408102154869648800813162339473, 8.826171809610314934244672989252, 9.445434149175178190534837201998

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