Properties

Label 4-192e2-1.1-c8e2-0-5
Degree $4$
Conductor $36864$
Sign $1$
Analytic cond. $6117.85$
Root an. cond. $8.84402$
Motivic weight $8$
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  − 90·3-s − 3.50e3·7-s + 1.53e3·9-s − 5.14e4·13-s − 3.78e4·19-s + 3.15e5·21-s + 7.30e5·25-s + 4.51e5·27-s − 7.02e5·31-s − 2.67e6·37-s + 4.63e6·39-s + 7.05e6·43-s − 2.34e6·49-s + 3.40e6·57-s − 1.50e6·61-s − 5.38e6·63-s − 4.53e6·67-s + 5.53e7·73-s − 6.57e7·75-s − 4.59e7·79-s − 5.07e7·81-s + 1.80e8·91-s + 6.32e7·93-s + 2.94e8·97-s − 3.32e8·103-s + 2.19e8·109-s + 2.40e8·111-s + ⋯
L(s)  = 1  − 1.11·3-s − 1.45·7-s + 0.234·9-s − 1.80·13-s − 0.290·19-s + 1.61·21-s + 1.87·25-s + 0.850·27-s − 0.761·31-s − 1.42·37-s + 2.00·39-s + 2.06·43-s − 0.406·49-s + 0.322·57-s − 0.108·61-s − 0.341·63-s − 0.225·67-s + 1.94·73-s − 2.07·75-s − 1.18·79-s − 1.17·81-s + 2.62·91-s + 0.845·93-s + 3.32·97-s − 2.95·103-s + 1.55·109-s + 1.58·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36864\)    =    \(2^{12} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(6117.85\)
Root analytic conductor: \(8.84402\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{192} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36864,\ (\ :4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.034069318\)
\(L(\frac12)\) \(\approx\) \(1.034069318\)
\(L(5)\) 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 \( 1 \)
3$C_2$ \( 1 + 10 p^{2} T + p^{8} T^{2} \)
good5$C_2^2$ \( 1 - 29234 p^{2} T^{2} + p^{16} T^{4} \)
7$C_2$ \( ( 1 + 250 p T + p^{8} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 380283362 T^{2} + p^{16} T^{4} \)
13$C_2$ \( ( 1 + 25730 T + p^{8} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 8342551298 T^{2} + p^{16} T^{4} \)
19$C_2$ \( ( 1 + 18938 T + p^{8} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 64711613182 T^{2} + p^{16} T^{4} \)
29$C_2^2$ \( 1 - 788066452322 T^{2} + p^{16} T^{4} \)
31$C_2$ \( ( 1 + 11338 p T + p^{8} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 1335170 T + p^{8} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 12452468931842 T^{2} + p^{16} T^{4} \)
43$C_2$ \( ( 1 - 3526150 T + p^{8} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30967680304898 T^{2} + p^{16} T^{4} \)
53$C_2^2$ \( 1 - 80936075395298 T^{2} + p^{16} T^{4} \)
59$C_2^2$ \( 1 - 105562517046242 T^{2} + p^{16} T^{4} \)
61$C_2$ \( ( 1 + 753602 T + p^{8} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2268890 T + p^{8} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 1001758688017922 T^{2} + p^{16} T^{4} \)
73$C_2$ \( ( 1 - 27672770 T + p^{8} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 22980982 T + p^{8} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 2352070843223138 T^{2} + p^{16} T^{4} \)
89$C_2^2$ \( 1 - 2600204109557762 T^{2} + p^{16} T^{4} \)
97$C_2$ \( ( 1 - 147271010 T + p^{8} T^{2} )^{2} \)
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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

−11.06671762039503512919171710739, −10.95199619443152712408454695640, −10.34310402186125021068822738825, −9.890147885638996593458816845502, −9.434310545929416109663100573315, −8.961821003411770063807393175033, −8.378888190616493291918597801539, −7.51095828558275356438528122437, −6.96386519361276280537838875669, −6.81696868690242026500493330561, −6.05913964667636834397211326158, −5.67999113124109079481047732626, −4.83521292397299381731124227231, −4.77739195772333346534271456346, −3.71250963817255084191982218853, −3.04960842830278460668915966046, −2.59263460572319220071525186713, −1.75769470888063035772433042056, −0.55928822855677271804628902786, −0.47441754253131420947644708206, 0.47441754253131420947644708206, 0.55928822855677271804628902786, 1.75769470888063035772433042056, 2.59263460572319220071525186713, 3.04960842830278460668915966046, 3.71250963817255084191982218853, 4.77739195772333346534271456346, 4.83521292397299381731124227231, 5.67999113124109079481047732626, 6.05913964667636834397211326158, 6.81696868690242026500493330561, 6.96386519361276280537838875669, 7.51095828558275356438528122437, 8.378888190616493291918597801539, 8.961821003411770063807393175033, 9.434310545929416109663100573315, 9.890147885638996593458816845502, 10.34310402186125021068822738825, 10.95199619443152712408454695640, 11.06671762039503512919171710739

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