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Properties

Label 4-1968e2-1.1-c1e2-0-1
Degree $4$
Conductor $3873024$
Sign $1$
Analytic cond. $246.947$
Root an. cond. $3.96415$
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  − 2·3-s − 4·5-s + 4·7-s + 3·9-s − 2·11-s − 4·13-s + 8·15-s − 2·17-s + 8·19-s − 8·21-s + 4·25-s − 4·27-s − 10·29-s + 10·31-s + 4·33-s − 16·35-s − 10·37-s + 8·39-s + 2·41-s + 14·43-s − 12·45-s + 6·47-s + 4·51-s + 8·53-s + 8·55-s − 16·57-s − 8·59-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 1.51·7-s + 9-s − 0.603·11-s − 1.10·13-s + 2.06·15-s − 0.485·17-s + 1.83·19-s − 1.74·21-s + 4/5·25-s − 0.769·27-s − 1.85·29-s + 1.79·31-s + 0.696·33-s − 2.70·35-s − 1.64·37-s + 1.28·39-s + 0.312·41-s + 2.13·43-s − 1.78·45-s + 0.875·47-s + 0.560·51-s + 1.09·53-s + 1.07·55-s − 2.11·57-s − 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3873024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3873024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3873024\)    =    \(2^{8} \cdot 3^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(246.947\)
Root analytic conductor: \(3.96415\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3873024,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9092263500\)
\(L(\frac12)\) \(\approx\) \(0.9092263500\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
41$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_m
7$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_q
11$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_f
13$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_bc
17$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_r
19$D_{4}$ \( 1 - 8 T + 36 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.19.ai_bk
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.23.a_bc
29$D_{4}$ \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_dd
31$D_{4}$ \( 1 - 10 T + 79 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_db
37$D_{4}$ \( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.37.k_dn
43$D_{4}$ \( 1 - 14 T + 127 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.43.ao_ex
47$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_cb
53$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_by
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_ew
61$D_{4}$ \( 1 + 6 T + 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_d
67$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_co
71$D_{4}$ \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_at
73$D_{4}$ \( 1 - 18 T + 195 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.73.as_hn
79$D_{4}$ \( 1 - 12 T + 66 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_co
83$D_{4}$ \( 1 + 12 T + 200 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_hs
89$D_{4}$ \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_ha
97$D_{4}$ \( 1 - 16 T + 208 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_ia
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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

−9.213124798527815034441136756945, −9.140141861986008310676910582867, −8.256333937173117152616825241680, −8.157863205017210435327539504409, −7.66223835160562841833025264104, −7.42910580998236155771801572649, −7.26652754361044593216854559853, −6.84287051992620055166423004886, −5.92297871681418627557248430314, −5.84095776411932685044828932761, −5.10642571811622716717596664598, −5.02342796658551854885496762258, −4.41819151269492038532468521692, −4.38814140410706975351820417564, −3.48073224248627951667355166619, −3.38439554426705482666427717283, −2.31351719061558114010365264093, −1.97742880186813042045329963211, −0.975707641763396122018176354344, −0.47112104466038903142811815631, 0.47112104466038903142811815631, 0.975707641763396122018176354344, 1.97742880186813042045329963211, 2.31351719061558114010365264093, 3.38439554426705482666427717283, 3.48073224248627951667355166619, 4.38814140410706975351820417564, 4.41819151269492038532468521692, 5.02342796658551854885496762258, 5.10642571811622716717596664598, 5.84095776411932685044828932761, 5.92297871681418627557248430314, 6.84287051992620055166423004886, 7.26652754361044593216854559853, 7.42910580998236155771801572649, 7.66223835160562841833025264104, 8.157863205017210435327539504409, 8.256333937173117152616825241680, 9.140141861986008310676910582867, 9.213124798527815034441136756945

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