Properties

Label 4-15e4-1.1-c3e2-0-9
Degree $4$
Conductor $50625$
Sign $1$
Analytic cond. $176.237$
Root an. cond. $3.64354$
Motivic weight $3$
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  + 2·2-s + 6·4-s + 26·7-s + 32·8-s − 28·11-s + 18·13-s + 52·14-s + 44·16-s + 68·17-s + 6·19-s − 56·22-s − 132·23-s + 36·26-s + 156·28-s − 92·29-s + 122·31-s + 120·32-s + 136·34-s − 284·37-s + 12·38-s − 392·41-s + 690·43-s − 168·44-s − 264·46-s + 620·47-s + 125·49-s + 108·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 3/4·4-s + 1.40·7-s + 1.41·8-s − 0.767·11-s + 0.384·13-s + 0.992·14-s + 0.687·16-s + 0.970·17-s + 0.0724·19-s − 0.542·22-s − 1.19·23-s + 0.271·26-s + 1.05·28-s − 0.589·29-s + 0.706·31-s + 0.662·32-s + 0.685·34-s − 1.26·37-s + 0.0512·38-s − 1.49·41-s + 2.44·43-s − 0.575·44-s − 0.846·46-s + 1.92·47-s + 0.364·49-s + 0.288·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50625\)    =    \(3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(176.237\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.635537035\)
\(L(\frac12)\) \(\approx\) \(5.635537035\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
good2$D_{4}$ \( 1 - p T - p T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 26 T + 551 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 28 T + 2554 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 18 T - 389 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 4 p T + 3382 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 13423 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 132 T + 25954 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 92 T + 35998 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 122 T + 48407 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 284 T + 110526 T^{2} + 284 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 392 T + 124882 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 690 T + 277735 T^{2} - 690 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 620 T + 284290 T^{2} - 620 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 16 p T + 462634 T^{2} - 16 p^{4} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 124 T + 336778 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 750 T + 535003 T^{2} - 750 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 358 T + 443567 T^{2} + 358 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 824 T + 877966 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 108 T + 779734 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 880 T + 693278 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 156 T + 449242 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 864 T + 1202578 T^{2} - 864 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 521 T + p^{3} T^{2} )^{2} \)
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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

−11.95851388148898864194797993291, −11.58503213193498472750046828568, −11.23468258661010407582143322923, −10.49623979524722635212629564958, −10.36104681288976294575126699450, −9.969584297037321457727190648333, −8.718405194840611930719111295067, −8.698487455767657844817834672950, −7.83562611549495458485057224085, −7.43577602898541656190030390795, −7.28844657601254109514013843367, −6.28694315425350630600039144592, −5.53725700464736748151346605983, −5.41575603912097788929603436192, −4.53663030746019483369748049728, −4.17703267214602182631512056044, −3.39149222797908716005704411854, −2.34466085850687691029983782745, −1.85549412255055055166707447739, −0.928512160325573181788221235424, 0.928512160325573181788221235424, 1.85549412255055055166707447739, 2.34466085850687691029983782745, 3.39149222797908716005704411854, 4.17703267214602182631512056044, 4.53663030746019483369748049728, 5.41575603912097788929603436192, 5.53725700464736748151346605983, 6.28694315425350630600039144592, 7.28844657601254109514013843367, 7.43577602898541656190030390795, 7.83562611549495458485057224085, 8.698487455767657844817834672950, 8.718405194840611930719111295067, 9.969584297037321457727190648333, 10.36104681288976294575126699450, 10.49623979524722635212629564958, 11.23468258661010407582143322923, 11.58503213193498472750046828568, 11.95851388148898864194797993291

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