Properties

Degree 4
Conductor $ 2^{4} $
Sign $1$
Motivic weight 17
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 5.88e3·3-s + 6.04e5·5-s + 2.53e7·7-s + 1.12e8·9-s + 1.25e9·11-s − 1.32e9·13-s − 3.55e9·15-s − 2.74e10·17-s + 1.01e11·19-s − 1.49e11·21-s + 1.34e11·23-s − 8.04e11·25-s − 1.88e12·27-s + 2.33e12·29-s + 2.78e11·31-s − 7.40e12·33-s + 1.53e13·35-s + 2.09e13·37-s + 7.76e12·39-s + 4.16e12·41-s − 1.11e14·43-s + 6.80e13·45-s + 1.96e14·47-s + 1.38e14·49-s + 1.61e14·51-s − 4.87e14·53-s + 7.60e14·55-s + ⋯
L(s)  = 1  − 0.517·3-s + 0.691·5-s + 1.66·7-s + 0.872·9-s + 1.77·11-s − 0.448·13-s − 0.357·15-s − 0.956·17-s + 1.36·19-s − 0.859·21-s + 0.358·23-s − 1.05·25-s − 1.28·27-s + 0.867·29-s + 0.0587·31-s − 0.916·33-s + 1.14·35-s + 0.979·37-s + 0.232·39-s + 0.0814·41-s − 1.45·43-s + 0.603·45-s + 1.20·47-s + 0.595·49-s + 0.494·51-s − 1.07·53-s + 1.22·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(17\)
character  :  induced by $\chi_{4} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 16,\ (\ :17/2, 17/2),\ 1)$
$L(9)$  $\approx$  $2.98068$
$L(\frac12)$  $\approx$  $2.98068$
$L(\frac{19}{2})$   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$D_{4}$ \( 1 + 1960 p T - 321646 p^{5} T^{2} + 1960 p^{18} T^{3} + p^{34} T^{4} \)
5$D_{4}$ \( 1 - 604044 T + 46794698206 p^{2} T^{2} - 604044 p^{17} T^{3} + p^{34} T^{4} \)
7$D_{4}$ \( 1 - 25350160 T + 10288992992430 p^{2} T^{2} - 25350160 p^{17} T^{3} + p^{34} T^{4} \)
11$D_{4}$ \( 1 - 114513480 p T + 7489675618956502 p^{2} T^{2} - 114513480 p^{18} T^{3} + p^{34} T^{4} \)
13$D_{4}$ \( 1 + 1320052580 T + 661181623540177494 p T^{2} + 1320052580 p^{17} T^{3} + p^{34} T^{4} \)
17$D_{4}$ \( 1 + 27498226140 T + \)\(18\!\cdots\!58\)\( T^{2} + 27498226140 p^{17} T^{3} + p^{34} T^{4} \)
19$D_{4}$ \( 1 - 101133633832 T + \)\(10\!\cdots\!34\)\( T^{2} - 101133633832 p^{17} T^{3} + p^{34} T^{4} \)
23$D_{4}$ \( 1 - 134767491120 T + \)\(21\!\cdots\!70\)\( T^{2} - 134767491120 p^{17} T^{3} + p^{34} T^{4} \)
29$D_{4}$ \( 1 - 2337155582652 T + \)\(94\!\cdots\!94\)\( T^{2} - 2337155582652 p^{17} T^{3} + p^{34} T^{4} \)
31$D_{4}$ \( 1 - 278836113472 T + \)\(39\!\cdots\!18\)\( T^{2} - 278836113472 p^{17} T^{3} + p^{34} T^{4} \)
37$D_{4}$ \( 1 - 20929802888140 T + \)\(93\!\cdots\!10\)\( T^{2} - 20929802888140 p^{17} T^{3} + p^{34} T^{4} \)
41$D_{4}$ \( 1 - 4166592315732 T + \)\(39\!\cdots\!18\)\( T^{2} - 4166592315732 p^{17} T^{3} + p^{34} T^{4} \)
43$D_{4}$ \( 1 + 111143148534440 T + \)\(13\!\cdots\!86\)\( T^{2} + 111143148534440 p^{17} T^{3} + p^{34} T^{4} \)
47$D_{4}$ \( 1 - 196772651157600 T + \)\(53\!\cdots\!18\)\( T^{2} - 196772651157600 p^{17} T^{3} + p^{34} T^{4} \)
53$D_{4}$ \( 1 + 487965122736660 T + \)\(41\!\cdots\!50\)\( T^{2} + 487965122736660 p^{17} T^{3} + p^{34} T^{4} \)
59$D_{4}$ \( 1 - 2835904197813624 T + \)\(45\!\cdots\!82\)\( T^{2} - 2835904197813624 p^{17} T^{3} + p^{34} T^{4} \)
61$D_{4}$ \( 1 + 67544034994436 T - \)\(59\!\cdots\!34\)\( T^{2} + 67544034994436 p^{17} T^{3} + p^{34} T^{4} \)
67$D_{4}$ \( 1 + 995171321546360 T + \)\(18\!\cdots\!70\)\( T^{2} + 995171321546360 p^{17} T^{3} + p^{34} T^{4} \)
71$D_{4}$ \( 1 - 3882245493215376 T + \)\(21\!\cdots\!26\)\( T^{2} - 3882245493215376 p^{17} T^{3} + p^{34} T^{4} \)
73$D_{4}$ \( 1 + 12746425881769580 T + \)\(13\!\cdots\!82\)\( T^{2} + 12746425881769580 p^{17} T^{3} + p^{34} T^{4} \)
79$D_{4}$ \( 1 + 14984271534065504 T + \)\(37\!\cdots\!22\)\( T^{2} + 14984271534065504 p^{17} T^{3} + p^{34} T^{4} \)
83$D_{4}$ \( 1 - 43899417809893800 T + \)\(13\!\cdots\!30\)\( T^{2} - 43899417809893800 p^{17} T^{3} + p^{34} T^{4} \)
89$D_{4}$ \( 1 - 51909007958846388 T + \)\(34\!\cdots\!94\)\( T^{2} - 51909007958846388 p^{17} T^{3} + p^{34} T^{4} \)
97$D_{4}$ \( 1 + 49281007789848380 T + \)\(91\!\cdots\!10\)\( T^{2} + 49281007789848380 p^{17} T^{3} + p^{34} T^{4} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.17200737763757841361731633284, −20.32924792156030931729415985185, −19.42914521055859307916552765402, −18.27783764260195604388851924085, −17.42272531046447011513111920059, −17.38258814186511238878605024469, −16.04523214199944693054209519596, −14.90895416535699676873902099127, −14.18912952905206708297251654682, −13.28890716299919749587649571602, −11.71454380078477834885769011083, −11.52409998933883939093132771380, −10.04523180569335592908663683829, −9.099802903730324013820154414837, −7.66199268442984915857746437293, −6.48035812582763516333469305742, −5.16076977806956282797043243787, −4.14802990103025714880467381097, −1.90081812298763992872817716425, −1.11495662939605257989208254741, 1.11495662939605257989208254741, 1.90081812298763992872817716425, 4.14802990103025714880467381097, 5.16076977806956282797043243787, 6.48035812582763516333469305742, 7.66199268442984915857746437293, 9.099802903730324013820154414837, 10.04523180569335592908663683829, 11.52409998933883939093132771380, 11.71454380078477834885769011083, 13.28890716299919749587649571602, 14.18912952905206708297251654682, 14.90895416535699676873902099127, 16.04523214199944693054209519596, 17.38258814186511238878605024469, 17.42272531046447011513111920059, 18.27783764260195604388851924085, 19.42914521055859307916552765402, 20.32924792156030931729415985185, 21.17200737763757841361731633284

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