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Properties

Label 4-48e2-1.1-c21e2-0-0
Degree $4$
Conductor $2304$
Sign $1$
Analytic cond. $17995.9$
Root an. cond. $11.5822$
Motivic weight $21$
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  − 1.18e5·3-s + 9.96e5·5-s − 6.79e8·7-s + 1.04e10·9-s − 2.19e11·11-s − 4.84e10·13-s − 1.17e11·15-s − 1.13e13·17-s − 1.19e13·19-s + 8.02e13·21-s + 1.46e14·23-s − 4.78e14·25-s − 8.23e14·27-s − 1.79e15·29-s − 1.11e16·31-s + 2.59e16·33-s − 6.77e14·35-s + 1.27e16·37-s + 5.72e15·39-s + 1.22e17·41-s − 2.88e17·43-s + 1.04e16·45-s − 8.37e17·47-s − 6.85e17·49-s + 1.33e18·51-s − 4.30e16·53-s − 2.19e17·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.0456·5-s − 0.909·7-s + 9-s − 2.55·11-s − 0.0975·13-s − 0.0527·15-s − 1.36·17-s − 0.447·19-s + 1.05·21-s + 0.737·23-s − 1.00·25-s − 0.769·27-s − 0.793·29-s − 2.44·31-s + 2.95·33-s − 0.0415·35-s + 0.435·37-s + 0.112·39-s + 1.43·41-s − 2.03·43-s + 0.0456·45-s − 2.32·47-s − 1.22·49-s + 1.57·51-s − 0.0337·53-s − 0.116·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(17995.9\)
Root analytic conductor: \(11.5822\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2304,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(0.1132720953\)
\(L(\frac12)\) \(\approx\) \(0.1132720953\)
\(L(\frac{23}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + p^{10} T )^{2} \)
good5$D_{4}$ \( 1 - 996876 T + 19189088648254 p^{2} T^{2} - 996876 p^{21} T^{3} + p^{42} T^{4} \)
7$D_{4}$ \( 1 + 97128016 p T + 23414651512456686 p^{2} T^{2} + 97128016 p^{22} T^{3} + p^{42} T^{4} \)
11$D_{4}$ \( 1 + 19988102088 p T + \)\(20\!\cdots\!54\)\( p^{2} T^{2} + 19988102088 p^{22} T^{3} + p^{42} T^{4} \)
13$D_{4}$ \( 1 + 48468909956 T - \)\(66\!\cdots\!82\)\( p T^{2} + 48468909956 p^{21} T^{3} + p^{42} T^{4} \)
17$D_{4}$ \( 1 + 666678178908 p T + \)\(56\!\cdots\!22\)\( p^{2} T^{2} + 666678178908 p^{22} T^{3} + p^{42} T^{4} \)
19$D_{4}$ \( 1 + 629504474296 p T + \)\(48\!\cdots\!38\)\( p^{2} T^{2} + 629504474296 p^{22} T^{3} + p^{42} T^{4} \)
23$D_{4}$ \( 1 - 146508390063504 T + \)\(70\!\cdots\!46\)\( T^{2} - 146508390063504 p^{21} T^{3} + p^{42} T^{4} \)
29$D_{4}$ \( 1 + 1798520043674052 T + \)\(75\!\cdots\!58\)\( T^{2} + 1798520043674052 p^{21} T^{3} + p^{42} T^{4} \)
31$D_{4}$ \( 1 + 11169107526944992 T + \)\(65\!\cdots\!62\)\( T^{2} + 11169107526944992 p^{21} T^{3} + p^{42} T^{4} \)
37$D_{4}$ \( 1 - 12736264858660012 T + \)\(11\!\cdots\!54\)\( T^{2} - 12736264858660012 p^{21} T^{3} + p^{42} T^{4} \)
41$D_{4}$ \( 1 - 122972020616468052 T + \)\(11\!\cdots\!02\)\( T^{2} - 122972020616468052 p^{21} T^{3} + p^{42} T^{4} \)
43$D_{4}$ \( 1 + 288455418162270040 T + \)\(60\!\cdots\!30\)\( T^{2} + 288455418162270040 p^{21} T^{3} + p^{42} T^{4} \)
47$D_{4}$ \( 1 + 837243745741596960 T + \)\(43\!\cdots\!10\)\( T^{2} + 837243745741596960 p^{21} T^{3} + p^{42} T^{4} \)
53$D_{4}$ \( 1 + 43007964012775764 T + \)\(30\!\cdots\!46\)\( T^{2} + 43007964012775764 p^{21} T^{3} + p^{42} T^{4} \)
59$D_{4}$ \( 1 - 3523823330903857224 T + \)\(22\!\cdots\!98\)\( T^{2} - 3523823330903857224 p^{21} T^{3} + p^{42} T^{4} \)
61$D_{4}$ \( 1 + 1779023128451013860 T + \)\(54\!\cdots\!38\)\( T^{2} + 1779023128451013860 p^{21} T^{3} + p^{42} T^{4} \)
67$D_{4}$ \( 1 - 16454068667621610296 T + \)\(48\!\cdots\!38\)\( T^{2} - 16454068667621610296 p^{21} T^{3} + p^{42} T^{4} \)
71$D_{4}$ \( 1 + 17379227131150420944 T + \)\(15\!\cdots\!26\)\( T^{2} + 17379227131150420944 p^{21} T^{3} + p^{42} T^{4} \)
73$D_{4}$ \( 1 - 50891146268473989076 T + \)\(15\!\cdots\!06\)\( T^{2} - 50891146268473989076 p^{21} T^{3} + p^{42} T^{4} \)
79$D_{4}$ \( 1 - 54055785594190591040 T + \)\(14\!\cdots\!58\)\( T^{2} - 54055785594190591040 p^{21} T^{3} + p^{42} T^{4} \)
83$D_{4}$ \( 1 + \)\(11\!\cdots\!88\)\( T + \)\(39\!\cdots\!78\)\( T^{2} + \)\(11\!\cdots\!88\)\( p^{21} T^{3} + p^{42} T^{4} \)
89$D_{4}$ \( 1 - \)\(22\!\cdots\!24\)\( T + \)\(17\!\cdots\!98\)\( T^{2} - \)\(22\!\cdots\!24\)\( p^{21} T^{3} + p^{42} T^{4} \)
97$D_{4}$ \( 1 + \)\(12\!\cdots\!36\)\( T + \)\(78\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!36\)\( p^{21} T^{3} + p^{42} T^{4} \)
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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

−11.40139510556744887121073178187, −11.25106965839400563506751315848, −10.72259924372898671469954847623, −10.18724851013766104433710254250, −9.608288033549874760122905239550, −9.155447495857736348764367623546, −8.014248946255929123888854970691, −7.889497233791262936800423986829, −6.80847092823784977905161709478, −6.75565724177711607703954500965, −5.80802489782648943315384068513, −5.37911758494585943760173414491, −4.98135854922922812283726945522, −4.28088647509277283177005830284, −3.45474248086505151295524693937, −2.91509137841013644736287329172, −2.01170375212765735221313472315, −1.83900229815880311317218577738, −0.51789548882294898357011320247, −0.13082036356501804553445518374, 0.13082036356501804553445518374, 0.51789548882294898357011320247, 1.83900229815880311317218577738, 2.01170375212765735221313472315, 2.91509137841013644736287329172, 3.45474248086505151295524693937, 4.28088647509277283177005830284, 4.98135854922922812283726945522, 5.37911758494585943760173414491, 5.80802489782648943315384068513, 6.75565724177711607703954500965, 6.80847092823784977905161709478, 7.889497233791262936800423986829, 8.014248946255929123888854970691, 9.155447495857736348764367623546, 9.608288033549874760122905239550, 10.18724851013766104433710254250, 10.72259924372898671469954847623, 11.25106965839400563506751315848, 11.40139510556744887121073178187

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