Properties

Label 4-933397-1.1-c1e2-0-0
Degree $4$
Conductor $933397$
Sign $1$
Analytic cond. $59.5142$
Root an. cond. $2.77750$
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 − 4-s − 6-s + 2·7-s + 8-s − 9-s + 11-s − 12-s + 5·13-s − 2·14-s − 16-s + 6·17-s + 18-s − 2·19-s + 2·21-s − 22-s − 4·23-s + 24-s − 6·25-s − 5·26-s − 2·28-s + 2·29-s + 7·31-s + 5·32-s + 33-s − 6·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s − 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.38·13-s − 0.534·14-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.436·21-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 6/5·25-s − 0.980·26-s − 0.377·28-s + 0.371·29-s + 1.25·31-s + 0.883·32-s + 0.174·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 933397 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933397 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(933397\)
Sign: $1$
Analytic conductor: \(59.5142\)
Root analytic conductor: \(2.77750\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{933397} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 933397,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.411263916\)
\(L(\frac12)\) \(\approx\) \(1.411263916\)
\(L(\frac{3}{2})\) 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)$
bad933397$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 526 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$D_{4}$ \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$D_{4}$ \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 18 T + 174 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 16 T + 182 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 154 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−12.0171331169, −11.7963519365, −11.4707545429, −10.8921625254, −10.4781704618, −10.0730221698, −9.81208439895, −9.33387316416, −8.82293730898, −8.55284862205, −8.24636130670, −7.97779732340, −7.63628012112, −6.80410189402, −6.45972119726, −5.95984090754, −5.52097517800, −4.91147210847, −4.48598047209, −3.85273721724, −3.43861983778, −2.95502318598, −1.97880477089, −1.54565543834, −0.637782812322, 0.637782812322, 1.54565543834, 1.97880477089, 2.95502318598, 3.43861983778, 3.85273721724, 4.48598047209, 4.91147210847, 5.52097517800, 5.95984090754, 6.45972119726, 6.80410189402, 7.63628012112, 7.97779732340, 8.24636130670, 8.55284862205, 8.82293730898, 9.33387316416, 9.81208439895, 10.0730221698, 10.4781704618, 10.8921625254, 11.4707545429, 11.7963519365, 12.0171331169

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