Properties

Degree 4
Conductor $ 2^{12} \cdot 5^{2} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s − 50·5-s − 52·7-s + 138·9-s + 560·11-s − 1.38e3·13-s + 600·15-s + 148·17-s − 1.00e3·19-s + 624·21-s + 2.45e3·23-s + 1.87e3·25-s − 4.50e3·27-s − 1.34e3·29-s + 2.24e3·31-s − 6.72e3·33-s + 2.60e3·35-s + 5.94e3·37-s + 1.66e4·39-s + 2.30e4·41-s + 1.76e4·43-s − 6.90e3·45-s + 2.90e3·47-s − 2.69e4·49-s − 1.77e3·51-s + 5.41e3·53-s − 2.80e4·55-s + ⋯
L(s)  = 1  − 0.769·3-s − 0.894·5-s − 0.401·7-s + 0.567·9-s + 1.39·11-s − 2.27·13-s + 0.688·15-s + 0.124·17-s − 0.635·19-s + 0.308·21-s + 0.966·23-s + 3/5·25-s − 1.18·27-s − 0.295·29-s + 0.420·31-s − 1.07·33-s + 0.358·35-s + 0.713·37-s + 1.75·39-s + 2.14·41-s + 1.45·43-s − 0.507·45-s + 0.192·47-s − 1.60·49-s − 0.0956·51-s + 0.264·53-s − 1.24·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(102400\)    =    \(2^{12} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{320} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 102400,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.314480318\)
\(L(\frac12)\)  \(\approx\)  \(1.314480318\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_1$ \( ( 1 + p^{2} T )^{2} \)
good3$D_{4}$ \( 1 + 4 p T + 2 p T^{2} + 4 p^{6} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 + 52 T + 29646 T^{2} + 52 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 560 T + 381926 T^{2} - 560 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 1388 T + 1149918 T^{2} + 1388 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 148 T - 795706 T^{2} - 148 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 1000 T + 4533462 T^{2} + 1000 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 2452 T + 6569198 T^{2} - 2452 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 1340 T - 8758306 T^{2} + 1340 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 2248 T + 57017022 T^{2} - 2248 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 5940 T + 123434318 T^{2} - 5940 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 23076 T + 352280470 T^{2} - 23076 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 17684 T + 312898614 T^{2} - 17684 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 2908 T + 56660030 T^{2} - 2908 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 5412 T + 693247822 T^{2} - 5412 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 62584 T + 2277965606 T^{2} - 62584 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 14108 T + 1110042462 T^{2} + 14108 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 85412 T + 4371910566 T^{2} + 85412 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 47208 T + 4011779662 T^{2} + 47208 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 924 p T + 4400780438 T^{2} + 924 p^{6} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 65904 T + 3994274078 T^{2} - 65904 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 108724 T + 10572459494 T^{2} - 108724 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 55020 T + 10818978262 T^{2} + 55020 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 147668 T + 11612429670 T^{2} - 147668 p^{5} T^{3} + p^{10} T^{4} \)
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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

−11.17039489028888081665863518308, −10.64634542512329349357174483015, −10.07232889159633244450729000728, −9.663812063200602081736876661948, −9.156794434106760203005833986952, −8.937013352479902503320234752145, −7.993014229724910133943923919022, −7.52779450447425965411115466863, −7.17810282774499158698847751502, −6.84725548450668614614820075779, −5.97664564326278683984393579847, −5.86244249313087855533492365488, −4.80800933717296692884006675698, −4.52682618883783459375324684044, −4.10755567594702296064159804403, −3.31713769161965297730074800483, −2.62819099335090632697236471852, −1.89234229947156225829846304425, −0.849361448395765761918563262542, −0.43098465786418360566905094916, 0.43098465786418360566905094916, 0.849361448395765761918563262542, 1.89234229947156225829846304425, 2.62819099335090632697236471852, 3.31713769161965297730074800483, 4.10755567594702296064159804403, 4.52682618883783459375324684044, 4.80800933717296692884006675698, 5.86244249313087855533492365488, 5.97664564326278683984393579847, 6.84725548450668614614820075779, 7.17810282774499158698847751502, 7.52779450447425965411115466863, 7.993014229724910133943923919022, 8.937013352479902503320234752145, 9.156794434106760203005833986952, 9.663812063200602081736876661948, 10.07232889159633244450729000728, 10.64634542512329349357174483015, 11.17039489028888081665863518308

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