Properties

Label 4-5e2-1.1-c5e2-0-0
Degree $4$
Conductor $25$
Sign $1$
Analytic cond. $0.643074$
Root an. cond. $0.895499$
Motivic weight $5$
Arithmetic yes
Rational yes
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

Degree: \(4\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $1$
Analytic conductor: \(0.643074\)
Root analytic conductor: \(0.895499\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8549440368\)
\(L(\frac12)\) \(\approx\) \(0.8549440368\)
\(L(\frac{7}{2})\) 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)$
bad5$C_2$ \( 1 + 18 p T + p^{5} T^{2} \)
good2$C_2^2$ \( 1 - 5 p^{2} T^{2} + p^{10} T^{4} \)
3$C_2$ \( ( 1 - 8 p T + p^{5} T^{2} )( 1 + 8 p T + p^{5} T^{2} ) \)
7$C_2^2$ \( 1 - 30050 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 252 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 728330 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 2363810 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 + 220 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6946370 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 6930 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6752 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 56462470 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 198 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 293842250 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 347593490 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 802472090 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 24660 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 5698 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 795787610 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 53352 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 883886830 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 - 51920 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 4053674810 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 9990 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 6923133890 T^{2} + p^{10} T^{4} \)
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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

−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