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Properties

Label 4-504e2-1.1-c1e2-0-41
Degree $4$
Conductor $254016$
Sign $1$
Analytic cond. $16.1962$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 4·7-s + 4·17-s + 2·25-s + 16·35-s + 4·37-s + 4·41-s − 16·47-s + 9·49-s + 16·67-s − 8·79-s − 16·83-s + 16·85-s + 4·89-s + 4·101-s − 12·109-s + 16·119-s − 2·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1.78·5-s + 1.51·7-s + 0.970·17-s + 2/5·25-s + 2.70·35-s + 0.657·37-s + 0.624·41-s − 2.33·47-s + 9/7·49-s + 1.95·67-s − 0.900·79-s − 1.75·83-s + 1.73·85-s + 0.423·89-s + 0.398·101-s − 1.14·109-s + 1.46·119-s − 0.181·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(254016\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(16.1962\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 254016,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.117802608\)
\(L(\frac12)\) \(\approx\) \(3.117802608\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.11.a_c
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ae_w
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.23.a_ag
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ae_cs
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.53.a_k
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.aq_ha
71$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.71.a_acs
73$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.73.a_ck
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.i_be
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.q_ig
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.ae_bm
97$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.97.a_da
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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

−9.120289580421790604239583449857, −8.332308583782182682586427712127, −8.030070305076211776093139061171, −7.71845213837477328679800995039, −6.97040408208445726386901898790, −6.46601905556324814975732603387, −5.93741703340916592394160606431, −5.44405470681185031904321393903, −5.24876669811712429966130256363, −4.56570718188292946050568175819, −3.97997157010983292617980611011, −3.10437440203547891401062886212, −2.35019811508946280449786011982, −1.78448511676758006638625015945, −1.24796043263408953786134995645, 1.24796043263408953786134995645, 1.78448511676758006638625015945, 2.35019811508946280449786011982, 3.10437440203547891401062886212, 3.97997157010983292617980611011, 4.56570718188292946050568175819, 5.24876669811712429966130256363, 5.44405470681185031904321393903, 5.93741703340916592394160606431, 6.46601905556324814975732603387, 6.97040408208445726386901898790, 7.71845213837477328679800995039, 8.030070305076211776093139061171, 8.332308583782182682586427712127, 9.120289580421790604239583449857

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