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Properties

Label 4-10125-1.1-c1e2-0-1
Degree $4$
Conductor $10125$
Sign $-1$
Analytic cond. $0.645578$
Root an. cond. $0.896369$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 2·4-s − 5-s − 6-s − 2·7-s − 3·8-s − 2·9-s − 10-s − 3·11-s + 2·12-s − 2·13-s − 2·14-s + 15-s + 16-s + 2·17-s − 2·18-s + 2·20-s + 2·21-s − 3·22-s + 3·23-s + 3·24-s + 25-s − 2·26-s + 5·27-s + 4·28-s − 5·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 1.06·8-s − 2/3·9-s − 0.316·10-s − 0.904·11-s + 0.577·12-s − 0.554·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.471·18-s + 0.447·20-s + 0.436·21-s − 0.639·22-s + 0.625·23-s + 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.962·27-s + 0.755·28-s − 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10125\)    =    \(3^{4} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(0.645578\)
Root analytic conductor: \(0.896369\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 10125,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
5$C_1$ \( 1 + T \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 7 T + 39 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$D_{4}$ \( 1 - 7 T + 28 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 93 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 19 T + 204 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 3 T - 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 152 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−16.6861410218, −16.3760497689, −15.9343350918, −15.1318982731, −14.7819210182, −14.4180063241, −13.6812446364, −13.3316528766, −12.8501906517, −12.4285954325, −11.9756753389, −11.3290444512, −10.6750841087, −10.2724000199, −9.43523466436, −9.03273490501, −8.50811325761, −7.57607647790, −7.20716934511, −6.13809661872, −5.52945518990, −5.16392404126, −4.36696989136, −3.56490832057, −2.79166804286, 0, 2.79166804286, 3.56490832057, 4.36696989136, 5.16392404126, 5.52945518990, 6.13809661872, 7.20716934511, 7.57607647790, 8.50811325761, 9.03273490501, 9.43523466436, 10.2724000199, 10.6750841087, 11.3290444512, 11.9756753389, 12.4285954325, 12.8501906517, 13.3316528766, 13.6812446364, 14.4180063241, 14.7819210182, 15.1318982731, 15.9343350918, 16.3760497689, 16.6861410218

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