Properties

Degree $4$
Conductor $5531904$
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  + 6·3-s + 20·5-s + 27·9-s + 20·11-s + 104·13-s + 120·15-s + 116·17-s − 192·19-s − 28·23-s + 148·25-s + 108·27-s + 296·29-s + 104·31-s + 120·33-s − 248·37-s + 624·39-s + 20·41-s + 720·43-s + 540·45-s + 96·47-s + 696·51-s + 268·53-s + 400·55-s − 1.15e3·57-s + 616·59-s + 16·61-s + 2.08e3·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 9-s + 0.548·11-s + 2.21·13-s + 2.06·15-s + 1.65·17-s − 2.31·19-s − 0.253·23-s + 1.18·25-s + 0.769·27-s + 1.89·29-s + 0.602·31-s + 0.633·33-s − 1.10·37-s + 2.56·39-s + 0.0761·41-s + 2.55·43-s + 1.78·45-s + 0.297·47-s + 1.91·51-s + 0.694·53-s + 0.980·55-s − 2.67·57-s + 1.35·59-s + 0.0335·61-s + 3.96·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(15.99876877\)
\(L(\frac12)\) \(\approx\) \(15.99876877\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 4 p T + 252 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 20 T + 1610 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 8 p T + 5848 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 116 T + 9140 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 192 T + 21966 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 28 T + 22962 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 296 T + 62994 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 104 T + 57286 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 248 T + 112074 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 20 T + 42020 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 720 T + 268614 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 96 T + 84950 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 268 T + 56510 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 616 T + 403470 T^{2} - 616 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 16 T + 454008 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 144 T + 57558 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 988 T + 852210 T^{2} + 988 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 104 T + 459136 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 944 T + 1097470 T^{2} - 944 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 1016 T + 1388838 T^{2} + 1016 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 388 T + 1217732 T^{2} + 388 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 488 T + 1167280 T^{2} - 488 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.699544707395168223402234937493, −8.600765404686976113795867009581, −8.205983820919049588746541901251, −7.928510577862917017990712081965, −7.19204988097997004476853538552, −6.87104891129427542044626215979, −6.28097400128542236307579102505, −6.25803493189634388279731340192, −5.67810439745547154991695163031, −5.63040398975028906856935926153, −4.74146556658921449018453982204, −4.30668552921410014364603549587, −3.87576287695656902629154541931, −3.60405455283160713942887329704, −2.83758758913221218801404914398, −2.64912745837785602672346448803, −1.88005445508107322124373851079, −1.78156884882026628038037781722, −0.999518116360043025161050530163, −0.832018042557373744609120251695, 0.832018042557373744609120251695, 0.999518116360043025161050530163, 1.78156884882026628038037781722, 1.88005445508107322124373851079, 2.64912745837785602672346448803, 2.83758758913221218801404914398, 3.60405455283160713942887329704, 3.87576287695656902629154541931, 4.30668552921410014364603549587, 4.74146556658921449018453982204, 5.63040398975028906856935926153, 5.67810439745547154991695163031, 6.25803493189634388279731340192, 6.28097400128542236307579102505, 6.87104891129427542044626215979, 7.19204988097997004476853538552, 7.928510577862917017990712081965, 8.205983820919049588746541901251, 8.600765404686976113795867009581, 8.699544707395168223402234937493

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