Properties

Label 4-15065-1.1-c1e2-0-0
Degree $4$
Conductor $15065$
Sign $1$
Analytic cond. $0.960557$
Root an. cond. $0.989990$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 5-s + 6-s + 2·7-s + 8-s + 9-s + 10-s − 3·11-s − 3·13-s + 2·14-s + 15-s − 16-s − 17-s + 18-s − 6·19-s + 2·21-s − 3·22-s + 23-s + 24-s − 4·25-s − 3·26-s + 4·27-s + 29-s + 30-s − 31-s − 6·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.832·13-s + 0.534·14-s + 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.37·19-s + 0.436·21-s − 0.639·22-s + 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.588·26-s + 0.769·27-s + 0.185·29-s + 0.182·30-s − 0.179·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15065 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15065 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15065\)    =    \(5 \cdot 23 \cdot 131\)
Sign: $1$
Analytic conductor: \(0.960557\)
Root analytic conductor: \(0.989990\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15065,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.856595216\)
\(L(\frac12)\) \(\approx\) \(1.856595216\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
131$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 21 T + p T^{2} ) \)
good2$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$D_{4}$ \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
43$D_{4}$ \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$D_{4}$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T + 32 T^{2} + p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$D_{4}$ \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$D_{4}$ \( 1 - 17 T + 248 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
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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

−15.7958732973, −15.5581788988, −14.9376334084, −14.6125650362, −14.0236070994, −13.7890876744, −13.2700533048, −12.6786239931, −12.5310630481, −11.6778357290, −11.0905799377, −10.5132108608, −10.2624692610, −9.38367128345, −8.97307985111, −8.27488580249, −7.75265267990, −7.24163743293, −6.48303493982, −5.69768177964, −4.95224823441, −4.55784168394, −3.84903191454, −2.58483783931, −2.03053541052, 2.03053541052, 2.58483783931, 3.84903191454, 4.55784168394, 4.95224823441, 5.69768177964, 6.48303493982, 7.24163743293, 7.75265267990, 8.27488580249, 8.97307985111, 9.38367128345, 10.2624692610, 10.5132108608, 11.0905799377, 11.6778357290, 12.5310630481, 12.6786239931, 13.2700533048, 13.7890876744, 14.0236070994, 14.6125650362, 14.9376334084, 15.5581788988, 15.7958732973

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