Properties

Degree $4$
Conductor $4096$
Sign $1$
Motivic weight $19$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.79e4·3-s − 1.22e6·5-s − 8.85e7·7-s − 4.01e8·9-s − 7.16e9·11-s + 1.01e10·13-s + 3.42e10·15-s − 7.20e10·17-s − 3.12e12·19-s + 2.47e12·21-s + 1.47e13·23-s − 3.44e13·25-s + 1.16e13·27-s + 3.02e13·29-s + 1.23e14·31-s + 1.99e14·33-s + 1.08e14·35-s − 2.01e15·37-s − 2.82e14·39-s + 2.54e15·41-s − 5.63e15·43-s + 4.92e14·45-s + 2.19e16·47-s − 3.29e15·49-s + 2.01e15·51-s + 9.41e15·53-s + 8.78e15·55-s + ⋯
L(s)  = 1  − 0.818·3-s − 0.280·5-s − 0.829·7-s − 0.345·9-s − 0.916·11-s + 0.264·13-s + 0.229·15-s − 0.147·17-s − 2.21·19-s + 0.678·21-s + 1.70·23-s − 1.80·25-s + 0.295·27-s + 0.387·29-s + 0.838·31-s + 0.749·33-s + 0.232·35-s − 2.54·37-s − 0.216·39-s + 1.21·41-s − 1.70·43-s + 0.0969·45-s + 2.86·47-s − 0.289·49-s + 0.120·51-s + 0.392·53-s + 0.257·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4096\)    =    \(2^{12}\)
Sign: $1$
Motivic weight: \(19\)
Character: induced by $\chi_{64} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4096,\ (\ :19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{21}{2})\) 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 \( 1 \)
good3$D_{4}$ \( 1 + 9304 p T + 14570470 p^{4} T^{2} + 9304 p^{20} T^{3} + p^{38} T^{4} \)
5$D_{4}$ \( 1 + 245324 p T + 1437226145582 p^{2} T^{2} + 245324 p^{20} T^{3} + p^{38} T^{4} \)
7$D_{4}$ \( 1 + 88510512 T + 1589951031584546 p T^{2} + 88510512 p^{19} T^{3} + p^{38} T^{4} \)
11$D_{4}$ \( 1 + 7163787608 T + 6207054254616739618 p T^{2} + 7163787608 p^{19} T^{3} + p^{38} T^{4} \)
13$D_{4}$ \( 1 - 10126923604 T + \)\(22\!\cdots\!66\)\( p T^{2} - 10126923604 p^{19} T^{3} + p^{38} T^{4} \)
17$D_{4}$ \( 1 + 4237945820 p T + \)\(26\!\cdots\!54\)\( p^{2} T^{2} + 4237945820 p^{20} T^{3} + p^{38} T^{4} \)
19$D_{4}$ \( 1 + 164235814328 p T + \)\(54\!\cdots\!14\)\( T^{2} + 164235814328 p^{20} T^{3} + p^{38} T^{4} \)
23$D_{4}$ \( 1 - 14759207090288 T + \)\(68\!\cdots\!70\)\( p T^{2} - 14759207090288 p^{19} T^{3} + p^{38} T^{4} \)
29$D_{4}$ \( 1 - 30249539245044 T + \)\(11\!\cdots\!22\)\( T^{2} - 30249539245044 p^{19} T^{3} + p^{38} T^{4} \)
31$D_{4}$ \( 1 - 123389562777920 T + \)\(27\!\cdots\!42\)\( T^{2} - 123389562777920 p^{19} T^{3} + p^{38} T^{4} \)
37$D_{4}$ \( 1 + 2015393170174524 T + \)\(60\!\cdots\!70\)\( p T^{2} + 2015393170174524 p^{19} T^{3} + p^{38} T^{4} \)
41$D_{4}$ \( 1 - 2540784959504244 T + \)\(74\!\cdots\!06\)\( T^{2} - 2540784959504244 p^{19} T^{3} + p^{38} T^{4} \)
43$D_{4}$ \( 1 + 5633655093389464 T + \)\(29\!\cdots\!38\)\( T^{2} + 5633655093389464 p^{19} T^{3} + p^{38} T^{4} \)
47$D_{4}$ \( 1 - 21948339587130336 T + \)\(23\!\cdots\!90\)\( T^{2} - 21948339587130336 p^{19} T^{3} + p^{38} T^{4} \)
53$D_{4}$ \( 1 - 9418125066904676 T + \)\(29\!\cdots\!78\)\( T^{2} - 9418125066904676 p^{19} T^{3} + p^{38} T^{4} \)
59$D_{4}$ \( 1 - 98542449590407624 T + \)\(95\!\cdots\!22\)\( T^{2} - 98542449590407624 p^{19} T^{3} + p^{38} T^{4} \)
61$D_{4}$ \( 1 + 10292145377839820 T + \)\(11\!\cdots\!82\)\( T^{2} + 10292145377839820 p^{19} T^{3} + p^{38} T^{4} \)
67$D_{4}$ \( 1 - 75753628003984504 T + \)\(44\!\cdots\!10\)\( T^{2} - 75753628003984504 p^{19} T^{3} + p^{38} T^{4} \)
71$D_{4}$ \( 1 + 17407052566713776 T + \)\(27\!\cdots\!06\)\( T^{2} + 17407052566713776 p^{19} T^{3} + p^{38} T^{4} \)
73$D_{4}$ \( 1 + 857508255059832268 T + \)\(53\!\cdots\!30\)\( T^{2} + 857508255059832268 p^{19} T^{3} + p^{38} T^{4} \)
79$D_{4}$ \( 1 - 226291921444855072 T + \)\(21\!\cdots\!34\)\( T^{2} - 226291921444855072 p^{19} T^{3} + p^{38} T^{4} \)
83$D_{4}$ \( 1 - 767515701460985048 T + \)\(59\!\cdots\!70\)\( T^{2} - 767515701460985048 p^{19} T^{3} + p^{38} T^{4} \)
89$D_{4}$ \( 1 - 6092545894435174548 T + \)\(31\!\cdots\!94\)\( T^{2} - 6092545894435174548 p^{19} T^{3} + p^{38} T^{4} \)
97$D_{4}$ \( 1 - 1548148249522347076 T + \)\(66\!\cdots\!10\)\( T^{2} - 1548148249522347076 p^{19} T^{3} + p^{38} T^{4} \)
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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

−10.79883216965990291838865073525, −10.47455153486627674365912182269, −10.13656861929590377405473625328, −9.191279183536262912225523525348, −8.667742040193141247945997860803, −8.326665999890917032298499845086, −7.42221079588182255347808030892, −6.98297267199730590994431126005, −6.17111913045106492873807341632, −6.08511826082408160116640113988, −5.19964944543956427413143200528, −4.87415531631144541387804294354, −3.87957257274799068608341918374, −3.67082238261696962180260016890, −2.58899255239411414933578793451, −2.43915530436479768990556441921, −1.50624267300141399561273272788, −0.67929799869599848299107229818, 0, 0, 0.67929799869599848299107229818, 1.50624267300141399561273272788, 2.43915530436479768990556441921, 2.58899255239411414933578793451, 3.67082238261696962180260016890, 3.87957257274799068608341918374, 4.87415531631144541387804294354, 5.19964944543956427413143200528, 6.08511826082408160116640113988, 6.17111913045106492873807341632, 6.98297267199730590994431126005, 7.42221079588182255347808030892, 8.326665999890917032298499845086, 8.667742040193141247945997860803, 9.191279183536262912225523525348, 10.13656861929590377405473625328, 10.47455153486627674365912182269, 10.79883216965990291838865073525

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