Properties

Label 4-98e2-1.1-c9e2-0-6
Degree $4$
Conductor $9604$
Sign $1$
Analytic cond. $2547.57$
Root an. cond. $7.10447$
Motivic weight $9$
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  + 16·2-s − 6·3-s + 560·5-s − 96·6-s − 4.09e3·8-s + 1.96e4·9-s + 8.96e3·10-s + 5.41e4·11-s + 2.26e5·13-s − 3.36e3·15-s − 6.55e4·16-s + 6.26e3·17-s + 3.14e5·18-s + 2.57e5·19-s + 8.66e5·22-s + 2.66e5·23-s + 2.45e4·24-s + 1.95e6·25-s + 3.62e6·26-s − 3.54e5·27-s + 3.14e6·29-s − 5.37e4·30-s − 4.63e6·31-s − 3.24e5·33-s + 1.00e5·34-s + 1.19e7·37-s + 4.11e6·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.0427·3-s + 0.400·5-s − 0.0302·6-s − 0.353·8-s + 9-s + 0.283·10-s + 1.11·11-s + 2.19·13-s − 0.0171·15-s − 1/4·16-s + 0.0181·17-s + 0.707·18-s + 0.452·19-s + 0.788·22-s + 0.198·23-s + 0.0151·24-s + 25-s + 1.55·26-s − 0.128·27-s + 0.826·29-s − 0.0121·30-s − 0.901·31-s − 0.0476·33-s + 0.0128·34-s + 1.04·37-s + 0.320·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9604\)    =    \(2^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2547.57\)
Root analytic conductor: \(7.10447\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{98} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9604,\ (\ :9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(7.855550036\)
\(L(\frac12)\) \(\approx\) \(7.855550036\)
\(L(\frac{11}{2})\) 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)$
bad2$C_2$ \( 1 - p^{4} T + p^{8} T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 2 p T - 2183 p^{2} T^{2} + 2 p^{10} T^{3} + p^{18} T^{4} \)
5$C_2^2$ \( 1 - 112 p T - 65581 p^{2} T^{2} - 112 p^{10} T^{3} + p^{18} T^{4} \)
11$C_2^2$ \( 1 - 54152 T + 574491413 T^{2} - 54152 p^{9} T^{3} + p^{18} T^{4} \)
13$C_2$ \( ( 1 - 113172 T + p^{9} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6262 T - 118548663853 T^{2} - 6262 p^{9} T^{3} + p^{18} T^{4} \)
19$C_2^2$ \( 1 - 257078 T - 256598599695 T^{2} - 257078 p^{9} T^{3} + p^{18} T^{4} \)
23$C_2^2$ \( 1 - 266000 T - 1730396661463 T^{2} - 266000 p^{9} T^{3} + p^{18} T^{4} \)
29$C_2$ \( ( 1 - 1574714 T + p^{9} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 4637484 T - 4933364310415 T^{2} + 4637484 p^{9} T^{3} + p^{18} T^{4} \)
37$C_2^2$ \( 1 - 11946238 T + 12750862557567 T^{2} - 11946238 p^{9} T^{3} + p^{18} T^{4} \)
41$C_2$ \( ( 1 + 21909126 T + p^{9} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 27520592 T + p^{9} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 52927836 T + 1682225350540129 T^{2} - 52927836 p^{9} T^{3} + p^{18} T^{4} \)
53$C_2^2$ \( 1 + 16221222 T - 3036635548628849 T^{2} + 16221222 p^{9} T^{3} + p^{18} T^{4} \)
59$C_2^2$ \( 1 + 140509618 T + 11079956931850985 T^{2} + 140509618 p^{9} T^{3} + p^{18} T^{4} \)
61$C_2^2$ \( 1 + 202963560 T + 29500060595039459 T^{2} + 202963560 p^{9} T^{3} + p^{18} T^{4} \)
67$C_2^2$ \( 1 + 153734572 T - 3572215768271763 T^{2} + 153734572 p^{9} T^{3} + p^{18} T^{4} \)
71$C_2$ \( ( 1 - 3938816 p T + p^{9} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 404022830 T + 104362860452940987 T^{2} + 404022830 p^{9} T^{3} + p^{18} T^{4} \)
79$C_2^2$ \( 1 - 130689816 T - 102771767976504463 T^{2} - 130689816 p^{9} T^{3} + p^{18} T^{4} \)
83$C_2$ \( ( 1 + 420134014 T + p^{9} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 469542390 T - 129886347700573109 T^{2} + 469542390 p^{9} T^{3} + p^{18} T^{4} \)
97$C_2$ \( ( 1 - 872501690 T + p^{9} T^{2} )^{2} \)
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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

−12.64136787460267938547748565225, −11.95214811335793928032129252900, −11.31011507995322492547217654139, −10.83357129122771176205132868319, −10.38051857060496762497354333210, −9.615747192724575143436813226792, −8.917197362096695763340283393851, −8.892249268233763691779811404672, −7.929656467312993824015607817053, −7.14054095748414189535037043277, −6.65096300654346586011837829712, −5.96124378202503181516345761046, −5.71147121032160896762566811941, −4.53809755938214586134319490728, −4.32422307772218678009783588745, −3.50659656983412659066913517309, −3.04332746415443094223628027236, −1.81284687976489637622760287894, −1.24826624450371507782370930583, −0.76257220081181064266713950327, 0.76257220081181064266713950327, 1.24826624450371507782370930583, 1.81284687976489637622760287894, 3.04332746415443094223628027236, 3.50659656983412659066913517309, 4.32422307772218678009783588745, 4.53809755938214586134319490728, 5.71147121032160896762566811941, 5.96124378202503181516345761046, 6.65096300654346586011837829712, 7.14054095748414189535037043277, 7.929656467312993824015607817053, 8.892249268233763691779811404672, 8.917197362096695763340283393851, 9.615747192724575143436813226792, 10.38051857060496762497354333210, 10.83357129122771176205132868319, 11.31011507995322492547217654139, 11.95214811335793928032129252900, 12.64136787460267938547748565225

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