Properties

Degree 4
Conductor $ 2^{6} $
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(64\)    =    \(2^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(19\)
character  :  induced by $\chi_{8} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 64,\ (\ :19/2, 19/2),\ 1)\)
\(L(10)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{21}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 4. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 3.
$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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.57546330941451224324464100797, −16.06603715345740463686071880437, −14.79102071372153259194674119065, −14.64888542996517703733943344671, −13.30733167267753409188456694186, −12.86868176830742796829676590207, −11.59222335920846488912957148529, −11.35313567923760706239587197006, −10.38026576371035753164136215114, −9.624696113091132773015213100438, −8.221718538408028992811871784240, −7.87366023001499326793804705964, −6.32387363019394249336450281071, −5.82184434280896855177527926660, −4.84883083718064255412935240813, −3.97188059486045678904331918274, −2.35086668857613831103081000435, −1.77777517828738202765974306924, 0, 0, 1.77777517828738202765974306924, 2.35086668857613831103081000435, 3.97188059486045678904331918274, 4.84883083718064255412935240813, 5.82184434280896855177527926660, 6.32387363019394249336450281071, 7.87366023001499326793804705964, 8.221718538408028992811871784240, 9.624696113091132773015213100438, 10.38026576371035753164136215114, 11.35313567923760706239587197006, 11.59222335920846488912957148529, 12.86868176830742796829676590207, 13.30733167267753409188456694186, 14.64888542996517703733943344671, 14.79102071372153259194674119065, 16.06603715345740463686071880437, 16.57546330941451224324464100797

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