Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{8} $
Sign $1$
Motivic weight 3
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 − 6·5-s + 16·7-s − 8·8-s − 12·10-s − 12·11-s − 38·13-s + 32·14-s − 16·16-s − 252·17-s + 40·19-s − 24·22-s − 168·23-s + 125·25-s − 76·26-s − 30·29-s + 88·31-s − 504·34-s − 96·35-s + 508·37-s + 80·38-s + 48·40-s − 42·41-s + 52·43-s − 336·46-s + 96·47-s + 343·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.536·5-s + 0.863·7-s − 0.353·8-s − 0.379·10-s − 0.328·11-s − 0.810·13-s + 0.610·14-s − 1/4·16-s − 3.59·17-s + 0.482·19-s − 0.232·22-s − 1.52·23-s + 25-s − 0.573·26-s − 0.192·29-s + 0.509·31-s − 2.54·34-s − 0.463·35-s + 2.25·37-s + 0.341·38-s + 0.189·40-s − 0.159·41-s + 0.184·43-s − 1.07·46-s + 0.297·47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(26244\)    =    \(2^{2} \cdot 3^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{162} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 26244,\ (\ :3/2, 3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.84750\)
\(L(\frac12)\)  \(\approx\)  \(1.84750\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - p T + p^{2} T^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 6 T - 89 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 16 T - 87 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 12 T - 1187 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 38 T - 753 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 126 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 20 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 168 T + 16057 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 30 T - 23489 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 88 T - 22047 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 254 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 42 T - 67157 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 52 T - 76803 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 96 T - 94607 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 198 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 660 T + 230221 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 538 T + 62463 T^{2} - 538 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 884 T + 480693 T^{2} + 884 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 792 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 218 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 520 T - 222639 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 - 492 T - 329723 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 810 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 1154 T + 419043 T^{2} + 1154 p^{3} T^{3} + p^{6} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.03671759598096539007792648426, −12.05375373847283134425463339839, −11.71859973283057404540896298569, −11.21399557918409339836763926114, −10.91977891829827884120185507012, −10.24899224800968626929539845872, −9.582233457319540021251574062611, −8.923259017835169994119256918490, −8.611895914387897786114526314319, −7.87840553880654713249005694560, −7.51158424068889554486111825326, −6.53150459804077004334445757144, −6.47152271349912096464703352360, −5.29398596060414535235289714627, −4.91159470364095355892009719459, −4.15641517158807286216801301998, −4.03230660664965857454205448834, −2.38185036243216017988320611905, −2.38071108501237782494702519733, −0.55414160022227999658429454210, 0.55414160022227999658429454210, 2.38071108501237782494702519733, 2.38185036243216017988320611905, 4.03230660664965857454205448834, 4.15641517158807286216801301998, 4.91159470364095355892009719459, 5.29398596060414535235289714627, 6.47152271349912096464703352360, 6.53150459804077004334445757144, 7.51158424068889554486111825326, 7.87840553880654713249005694560, 8.611895914387897786114526314319, 8.923259017835169994119256918490, 9.582233457319540021251574062611, 10.24899224800968626929539845872, 10.91977891829827884120185507012, 11.21399557918409339836763926114, 11.71859973283057404540896298569, 12.05375373847283134425463339839, 13.03671759598096539007792648426

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