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 + 3·5-s + 27·9-s + 51·11-s − 61·13-s − 18·15-s − 24·17-s + 169·19-s + 192·23-s − 115·25-s − 108·27-s − 39·29-s − 92·31-s − 306·33-s − 173·37-s + 366·39-s − 174·41-s + 497·43-s + 81·45-s + 180·47-s + 144·51-s + 285·53-s + 153·55-s − 1.01e3·57-s − 1.26e3·59-s − 328·61-s − 183·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.268·5-s + 9-s + 1.39·11-s − 1.30·13-s − 0.309·15-s − 0.342·17-s + 2.04·19-s + 1.74·23-s − 0.919·25-s − 0.769·27-s − 0.249·29-s − 0.533·31-s − 1.61·33-s − 0.768·37-s + 1.50·39-s − 0.662·41-s + 1.76·43-s + 0.268·45-s + 0.558·47-s + 0.395·51-s + 0.738·53-s + 0.375·55-s − 2.35·57-s − 2.80·59-s − 0.688·61-s − 0.349·65-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

\(L(2)\) \(\approx\) \(3.244360948\)
\(L(\frac12)\) \(\approx\) \(3.244360948\)
\(L(\frac{5}{2})\) 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$D_{4}$ \( 1 - 3 T + 124 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 51 T + 3184 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 61 T + 2118 T^{2} + 61 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 24 T + 1762 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 169 T + 20730 T^{2} - 169 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 96 T + p^{3} T^{2} )^{2} \)
29$D_{4}$ \( 1 + 39 T + 12094 T^{2} + 39 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 92 T + 48873 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 173 T + 62490 T^{2} + 173 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 174 T - 2846 T^{2} + 174 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 497 T + 174468 T^{2} - 497 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 180 T + 182914 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 285 T + 224566 T^{2} - 285 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 1269 T + 13766 p T^{2} + 1269 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 328 T + 314646 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 875 T + 782544 T^{2} - 875 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1404 T + 1077298 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1361 T + 1184556 T^{2} - 1361 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 182 T + 992307 T^{2} - 182 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 399 T + 946240 T^{2} + 399 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 822 T + 1516786 T^{2} + 822 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 841 T + 1945608 T^{2} + 841 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.103464380242617553511098400996, −8.515776966569671419793689465071, −7.83108889368526994990910407245, −7.54629153979818213215710048868, −7.09598964342515668281207126588, −7.06807268117595668821653741597, −6.35297706838601530476770412716, −6.25995647612891808886014689614, −5.47227013401897378005324302583, −5.42860290427695971804113790803, −4.86869187155447961250320973521, −4.71698154816232514611088165826, −3.90041283273099290310489182724, −3.71352353418520223317620954081, −3.09425955349054042103898303139, −2.54251591136882356422614091910, −1.84786725164573759329089512512, −1.47748345174797004057789524296, −0.64608099154644419861153798470, −0.63255798946810150526164685050, 0.63255798946810150526164685050, 0.64608099154644419861153798470, 1.47748345174797004057789524296, 1.84786725164573759329089512512, 2.54251591136882356422614091910, 3.09425955349054042103898303139, 3.71352353418520223317620954081, 3.90041283273099290310489182724, 4.71698154816232514611088165826, 4.86869187155447961250320973521, 5.42860290427695971804113790803, 5.47227013401897378005324302583, 6.25995647612891808886014689614, 6.35297706838601530476770412716, 7.06807268117595668821653741597, 7.09598964342515668281207126588, 7.54629153979818213215710048868, 7.83108889368526994990910407245, 8.515776966569671419793689465071, 9.103464380242617553511098400996

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