Properties

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

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 20·4-s − 90·5-s + 90·9-s + 504·11-s − 624·16-s − 440·19-s − 1.80e3·20-s + 4.97e3·25-s − 1.38e4·29-s + 1.35e4·31-s + 1.80e3·36-s − 396·41-s + 1.00e4·44-s − 8.10e3·45-s + 3.00e4·49-s − 4.53e4·55-s − 4.93e4·59-s − 1.13e4·61-s − 3.29e4·64-s + 1.06e5·71-s − 8.80e3·76-s + 1.03e5·79-s + 5.61e4·80-s − 5.09e4·81-s − 1.99e4·89-s + 3.96e4·95-s + 4.53e4·99-s + ⋯
L(s)  = 1  + 5/8·4-s − 1.60·5-s + 0.370·9-s + 1.25·11-s − 0.609·16-s − 0.279·19-s − 1.00·20-s + 1.59·25-s − 3.06·29-s + 2.52·31-s + 0.231·36-s − 0.0367·41-s + 0.784·44-s − 0.596·45-s + 1.78·49-s − 2.02·55-s − 1.84·59-s − 0.392·61-s − 1.00·64-s + 2.51·71-s − 0.174·76-s + 1.87·79-s + 0.981·80-s − 0.862·81-s − 0.267·89-s + 0.450·95-s + 0.465·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(25\)    =    \(5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{5} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 25,\ (\ :5/2, 5/2),\ 1)$
$L(3)$  $\approx$  $0.854944$
$L(\frac12)$  $\approx$  $0.854944$
$L(\frac{7}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 4. If $p = 5$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_2$ \( 1 + 18 p T + p^{5} T^{2} \)
good2$V_4$ \( 1 - 5 p^{2} T^{2} + p^{10} T^{4} \)
3$C_2$ \( ( 1 - 8 p T + p^{5} T^{2} )( 1 + 8 p T + p^{5} T^{2} ) \)
7$V_4$ \( 1 - 30050 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 252 T + p^{5} T^{2} )^{2} \)
13$V_4$ \( 1 - 728330 T^{2} + p^{10} T^{4} \)
17$V_4$ \( 1 - 2363810 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 + 220 T + p^{5} T^{2} )^{2} \)
23$V_4$ \( 1 - 6946370 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 6930 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6752 T + p^{5} T^{2} )^{2} \)
37$V_4$ \( 1 + 56462470 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 198 T + p^{5} T^{2} )^{2} \)
43$V_4$ \( 1 - 293842250 T^{2} + p^{10} T^{4} \)
47$V_4$ \( 1 - 347593490 T^{2} + p^{10} T^{4} \)
53$V_4$ \( 1 - 802472090 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 24660 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 5698 T + p^{5} T^{2} )^{2} \)
67$V_4$ \( 1 - 795787610 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 53352 T + p^{5} T^{2} )^{2} \)
73$V_4$ \( 1 + 883886830 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 - 51920 T + p^{5} T^{2} )^{2} \)
83$V_4$ \( 1 - 4053674810 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 9990 T + p^{5} T^{2} )^{2} \)
97$V_4$ \( 1 - 6923133890 T^{2} + p^{10} T^{4} \)
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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

−23.78022987484324488734829947865, −22.80613211102536782471174831139, −22.52166057003992423437663482243, −21.35581585461900188282141236080, −20.36999132347011813021972513358, −19.96105957672780268456689498558, −19.11732653065298533820811683384, −18.56236613057734487804330711583, −17.09272587952248977110949467303, −16.48826307049993962111149144423, −15.33533538841285802399005921429, −15.22209378099097969216612692439, −13.79103441325280814819768052363, −12.38331049598517745598077461804, −11.66134790726682541804104153830, −10.93869529992611551262793030092, −9.223229959911089700441868929878, −7.80950001073798191226471943289, −6.71631094869763221321108390169, −4.06350745703308212762258906706, 4.06350745703308212762258906706, 6.71631094869763221321108390169, 7.80950001073798191226471943289, 9.223229959911089700441868929878, 10.93869529992611551262793030092, 11.66134790726682541804104153830, 12.38331049598517745598077461804, 13.79103441325280814819768052363, 15.22209378099097969216612692439, 15.33533538841285802399005921429, 16.48826307049993962111149144423, 17.09272587952248977110949467303, 18.56236613057734487804330711583, 19.11732653065298533820811683384, 19.96105957672780268456689498558, 20.36999132347011813021972513358, 21.35581585461900188282141236080, 22.52166057003992423437663482243, 22.80613211102536782471174831139, 23.78022987484324488734829947865

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