Properties

Label 4-60617-1.1-c1e2-0-0
Degree $4$
Conductor $60617$
Sign $-1$
Analytic cond. $3.86499$
Root an. cond. $1.40212$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 4·3-s + 4·4-s − 3·5-s + 12·6-s − 4·7-s − 3·8-s + 7·9-s + 9·10-s − 5·11-s − 16·12-s − 10·13-s + 12·14-s + 12·15-s + 3·16-s − 4·17-s − 21·18-s − 5·19-s − 12·20-s + 16·21-s + 15·22-s − 23-s + 12·24-s − 25-s + 30·26-s − 4·27-s − 16·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.30·3-s + 2·4-s − 1.34·5-s + 4.89·6-s − 1.51·7-s − 1.06·8-s + 7/3·9-s + 2.84·10-s − 1.50·11-s − 4.61·12-s − 2.77·13-s + 3.20·14-s + 3.09·15-s + 3/4·16-s − 0.970·17-s − 4.94·18-s − 1.14·19-s − 2.68·20-s + 3.49·21-s + 3.19·22-s − 0.208·23-s + 2.44·24-s − 1/5·25-s + 5.88·26-s − 0.769·27-s − 3.02·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60617 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60617 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(60617\)
Sign: $-1$
Analytic conductor: \(3.86499\)
Root analytic conductor: \(1.40212\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((4,\ 60617,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad60617$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 27 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 10 T + 48 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + 31 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 121 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T - 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 15 T + 137 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 3 T + 83 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 20 T + 251 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - T - 23 T^{2} - p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 15 T + 223 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T - 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−15.6527887188, −15.0455138939, −14.7987652432, −13.6892547393, −13.0623737234, −12.5145194732, −12.3499387553, −11.9168576794, −11.5345592563, −10.9374235202, −10.4543440646, −10.2684310105, −9.83171468767, −9.41838441494, −8.68405694507, −8.17075921290, −7.60458697097, −7.28907689222, −6.61850081930, −6.32982059017, −5.45916821821, −4.94470650065, −4.45279804683, −3.24211861617, −2.31208914934, 0, 0, 0, 2.31208914934, 3.24211861617, 4.45279804683, 4.94470650065, 5.45916821821, 6.32982059017, 6.61850081930, 7.28907689222, 7.60458697097, 8.17075921290, 8.68405694507, 9.41838441494, 9.83171468767, 10.2684310105, 10.4543440646, 10.9374235202, 11.5345592563, 11.9168576794, 12.3499387553, 12.5145194732, 13.0623737234, 13.6892547393, 14.7987652432, 15.0455138939, 15.6527887188

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