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Properties

Label 4-6848e2-1.1-c1e2-0-11
Degree $4$
Conductor $46895104$
Sign $1$
Analytic cond. $2990.07$
Root an. cond. $7.39469$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s + 2·7-s + 2·11-s − 2·13-s + 8·15-s + 10·17-s + 4·19-s − 4·21-s + 5·25-s + 2·27-s − 10·29-s + 4·31-s − 4·33-s − 8·35-s + 8·37-s + 4·39-s + 6·41-s − 18·43-s − 8·49-s − 20·51-s − 8·53-s − 8·55-s − 8·57-s + 10·59-s − 2·61-s + 8·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 0.755·7-s + 0.603·11-s − 0.554·13-s + 2.06·15-s + 2.42·17-s + 0.917·19-s − 0.872·21-s + 25-s + 0.384·27-s − 1.85·29-s + 0.718·31-s − 0.696·33-s − 1.35·35-s + 1.31·37-s + 0.640·39-s + 0.937·41-s − 2.74·43-s − 8/7·49-s − 2.80·51-s − 1.09·53-s − 1.07·55-s − 1.05·57-s + 1.30·59-s − 0.256·61-s + 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46895104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46895104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(46895104\)    =    \(2^{12} \cdot 107^{2}\)
Sign: $1$
Analytic conductor: \(2990.07\)
Root analytic conductor: \(7.39469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 46895104,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
107$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_l
7$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_m
11$D_{4}$ \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_u
13$D_{4}$ \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_y
17$D_{4}$ \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.17.ak_ce
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \) 2.23.a_br
29$D_{4}$ \( 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_dc
31$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_s
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_br
43$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.43.s_gl
47$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \) 2.47.a_dn
53$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_eg
59$D_{4}$ \( 1 - 10 T + 131 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_fb
61$D_{4}$ \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_bw
67$D_{4}$ \( 1 - 10 T + 111 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_eh
71$D_{4}$ \( 1 + 6 T + 4 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_e
73$D_{4}$ \( 1 + 10 T + 168 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.73.k_gm
79$D_{4}$ \( 1 - 4 T - 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_add
83$D_{4}$ \( 1 + 18 T + 172 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.83.s_gq
89$D_{4}$ \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_db
97$D_{4}$ \( 1 - 6 T - 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_abo
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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

−7.79521152125264141852779316343, −7.63500321630540778420608014165, −7.13031760330101922150711486201, −6.85116732522497662868003028362, −6.27755941927904101379843659082, −6.05720491216796758604867534966, −5.44268490063648836120772453322, −5.40922917169332938068868709702, −5.03970477372935698954825483551, −4.61871154178757053325081055850, −4.16862900327118290271297280307, −3.83048199210492718549615310337, −3.39964282152106966391355665851, −3.20254229874599648206044170353, −2.64458259926336896875055675743, −1.87927350582481705161938925620, −1.20657897148278300127213735097, −1.07844534582920694465019689651, 0, 0, 1.07844534582920694465019689651, 1.20657897148278300127213735097, 1.87927350582481705161938925620, 2.64458259926336896875055675743, 3.20254229874599648206044170353, 3.39964282152106966391355665851, 3.83048199210492718549615310337, 4.16862900327118290271297280307, 4.61871154178757053325081055850, 5.03970477372935698954825483551, 5.40922917169332938068868709702, 5.44268490063648836120772453322, 6.05720491216796758604867534966, 6.27755941927904101379843659082, 6.85116732522497662868003028362, 7.13031760330101922150711486201, 7.63500321630540778420608014165, 7.79521152125264141852779316343

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