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Properties

Label 4-304e2-1.1-c1e2-0-9
Degree $4$
Conductor $92416$
Sign $1$
Analytic cond. $5.89252$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4·5-s + 3·9-s − 6·11-s − 2·13-s + 4·15-s − 2·17-s − 19-s + 6·23-s + 5·25-s + 8·27-s + 4·29-s + 20·31-s − 6·33-s + 4·37-s − 2·39-s − 9·41-s − 4·43-s + 12·45-s − 12·47-s − 14·49-s − 2·51-s + 2·53-s − 24·55-s − 57-s − 59-s + 8·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s + 9-s − 1.80·11-s − 0.554·13-s + 1.03·15-s − 0.485·17-s − 0.229·19-s + 1.25·23-s + 25-s + 1.53·27-s + 0.742·29-s + 3.59·31-s − 1.04·33-s + 0.657·37-s − 0.320·39-s − 1.40·41-s − 0.609·43-s + 1.78·45-s − 1.75·47-s − 2·49-s − 0.280·51-s + 0.274·53-s − 3.23·55-s − 0.132·57-s − 0.130·59-s + 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(92416\)    =    \(2^{8} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5.89252\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 92416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.466804212\)
\(L(\frac12)\) \(\approx\) \(2.466804212\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_l
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_an
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_an
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_abb
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_dt
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_abx
59$C_2^2$ \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) 2.59.b_acg
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_d
67$C_2^2$ \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.67.aj_o
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2^2$ \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.73.aj_i
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.e_acl
83$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.83.ak_hj
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_jb
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) 2.97.b_ads
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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

−11.76258072278365347564656193090, −11.72198242400185260630345459325, −10.62440029208520472919420562422, −10.39000189893979350036112191182, −10.12687790120778233184382470633, −9.636470324861114824405229001086, −9.412106672404616378564180251946, −8.534495941313028896513287988989, −8.079281494323192069601970622651, −7.972884941498347770618588187395, −6.82249317969547774926746422880, −6.64716098032331139238775547717, −6.23671848197268480345670693083, −5.25777917572658162765240638035, −4.76808597674885143024624711630, −4.76325260119392571148515262340, −3.30921564315505916031443927667, −2.62006520481166113612200825421, −2.34711947397844923830619801392, −1.29217626946479508523343033710, 1.29217626946479508523343033710, 2.34711947397844923830619801392, 2.62006520481166113612200825421, 3.30921564315505916031443927667, 4.76325260119392571148515262340, 4.76808597674885143024624711630, 5.25777917572658162765240638035, 6.23671848197268480345670693083, 6.64716098032331139238775547717, 6.82249317969547774926746422880, 7.972884941498347770618588187395, 8.079281494323192069601970622651, 8.534495941313028896513287988989, 9.412106672404616378564180251946, 9.636470324861114824405229001086, 10.12687790120778233184382470633, 10.39000189893979350036112191182, 10.62440029208520472919420562422, 11.72198242400185260630345459325, 11.76258072278365347564656193090

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