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Properties

Label 4-160e2-1.1-c1e2-0-11
Degree $4$
Conductor $25600$
Sign $1$
Analytic cond. $1.63227$
Root an. cond. $1.13031$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 4·5-s + 4·7-s + 8·9-s − 2·13-s − 16·15-s − 10·17-s + 8·19-s + 16·21-s + 4·23-s + 11·25-s + 12·27-s − 16·35-s + 2·37-s − 8·39-s − 12·43-s − 32·45-s − 4·47-s + 8·49-s − 40·51-s − 14·53-s + 32·57-s + 8·59-s − 8·61-s + 32·63-s + 8·65-s − 20·67-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.78·5-s + 1.51·7-s + 8/3·9-s − 0.554·13-s − 4.13·15-s − 2.42·17-s + 1.83·19-s + 3.49·21-s + 0.834·23-s + 11/5·25-s + 2.30·27-s − 2.70·35-s + 0.328·37-s − 1.28·39-s − 1.82·43-s − 4.77·45-s − 0.583·47-s + 8/7·49-s − 5.60·51-s − 1.92·53-s + 4.23·57-s + 1.04·59-s − 1.02·61-s + 4.03·63-s + 0.992·65-s − 2.44·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25600\)    =    \(2^{10} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1.63227\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.079274822\)
\(L(\frac12)\) \(\approx\) \(2.079274822\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.3.ae_i
7$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_i
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.c_c
17$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.k_by
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_i
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_c
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_cu
47$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_i
53$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.53.o_du
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.61.i_fi
67$C_2^2$ \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.67.u_hs
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_s
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_i
89$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.89.a_agw
97$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_s
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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

−13.52948900660978997700739454452, −12.71374801099312588499214377472, −12.07023270002773118411229294502, −11.67769579780215213176866036881, −11.02035611823193949200370460922, −10.98230605233472102033642787198, −9.939661053809370859920349707089, −9.126987559540030977544371724928, −9.010490516613839272564350717719, −8.413988397954181114823838919408, −7.929833937490584685691239283390, −7.79759963680671264523446257947, −7.14425004370225237975692630081, −6.65685970773035818937017623377, −4.89861537428938523849156835685, −4.80279718484695034016716814816, −4.04291414657135911424569089257, −3.24165158360942039221965694743, −2.79192753768080197357103162731, −1.72813720098024007817035789707, 1.72813720098024007817035789707, 2.79192753768080197357103162731, 3.24165158360942039221965694743, 4.04291414657135911424569089257, 4.80279718484695034016716814816, 4.89861537428938523849156835685, 6.65685970773035818937017623377, 7.14425004370225237975692630081, 7.79759963680671264523446257947, 7.929833937490584685691239283390, 8.413988397954181114823838919408, 9.010490516613839272564350717719, 9.126987559540030977544371724928, 9.939661053809370859920349707089, 10.98230605233472102033642787198, 11.02035611823193949200370460922, 11.67769579780215213176866036881, 12.07023270002773118411229294502, 12.71374801099312588499214377472, 13.52948900660978997700739454452

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