Properties

Label 4-306e2-1.1-c1e2-0-11
Degree $4$
Conductor $93636$
Sign $1$
Analytic cond. $5.97031$
Root an. cond. $1.56314$
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  + 2·2-s + 3·4-s + 4·7-s + 4·8-s + 4·13-s + 8·14-s + 5·16-s − 2·17-s + 4·19-s − 12·23-s − 4·25-s + 8·26-s + 12·28-s + 4·31-s + 6·32-s − 4·34-s − 8·37-s + 8·38-s − 12·41-s − 8·43-s − 24·46-s + 4·49-s − 8·50-s + 12·52-s + 12·53-s + 16·56-s − 12·59-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.51·7-s + 1.41·8-s + 1.10·13-s + 2.13·14-s + 5/4·16-s − 0.485·17-s + 0.917·19-s − 2.50·23-s − 4/5·25-s + 1.56·26-s + 2.26·28-s + 0.718·31-s + 1.06·32-s − 0.685·34-s − 1.31·37-s + 1.29·38-s − 1.87·41-s − 1.21·43-s − 3.53·46-s + 4/7·49-s − 1.13·50-s + 1.66·52-s + 1.64·53-s + 2.13·56-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93636\)    =    \(2^{2} \cdot 3^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(5.97031\)
Root analytic conductor: \(1.56314\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 93636,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.266352391\)
\(L(\frac12)\) \(\approx\) \(4.266352391\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 12 T + 76 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 84 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 4 T + 108 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 178 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 20 T + 270 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
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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

−11.84113585983832538063280761474, −11.77512466641311665591739125176, −11.30194701010089363838751784007, −10.64898817665341362940921388687, −10.19002059170127308076586554775, −9.992046172226905601459623132759, −8.833751583248263675224159267907, −8.613410583582073825287345107565, −7.86904860361687512736336854243, −7.74357543422960419268710618884, −6.98757685114717043892086454957, −6.33857291204919403842417010961, −5.89384987279117372246307038874, −5.45364476254799833238769270848, −4.68761657311222396212359772937, −4.47768029060519589093036931803, −3.64098565511974959530421417733, −3.23989158761777289249697129132, −1.91924163491180388408358818366, −1.71772058640558074185347431055, 1.71772058640558074185347431055, 1.91924163491180388408358818366, 3.23989158761777289249697129132, 3.64098565511974959530421417733, 4.47768029060519589093036931803, 4.68761657311222396212359772937, 5.45364476254799833238769270848, 5.89384987279117372246307038874, 6.33857291204919403842417010961, 6.98757685114717043892086454957, 7.74357543422960419268710618884, 7.86904860361687512736336854243, 8.613410583582073825287345107565, 8.833751583248263675224159267907, 9.992046172226905601459623132759, 10.19002059170127308076586554775, 10.64898817665341362940921388687, 11.30194701010089363838751784007, 11.77512466641311665591739125176, 11.84113585983832538063280761474

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