Properties

Label 4-177e2-1.1-c1e2-0-0
Degree $4$
Conductor $31329$
Sign $1$
Analytic cond. $1.99756$
Root an. cond. $1.18884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s − 2·4-s + 2·5-s + 2·6-s + 7-s − 3·8-s + 3·9-s + 2·10-s + 4·11-s − 4·12-s − 2·13-s + 14-s + 4·15-s + 16-s + 17-s + 3·18-s − 5·19-s − 4·20-s + 2·21-s + 4·22-s + 3·23-s − 6·24-s − 7·25-s − 2·26-s + 4·27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 4-s + 0.894·5-s + 0.816·6-s + 0.377·7-s − 1.06·8-s + 9-s + 0.632·10-s + 1.20·11-s − 1.15·12-s − 0.554·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s + 0.242·17-s + 0.707·18-s − 1.14·19-s − 0.894·20-s + 0.436·21-s + 0.852·22-s + 0.625·23-s − 1.22·24-s − 7/5·25-s − 0.392·26-s + 0.769·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31329\)    =    \(3^{2} \cdot 59^{2}\)
Sign: $1$
Analytic conductor: \(1.99756\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 31329,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.368557994\)
\(L(\frac12)\) \(\approx\) \(2.368557994\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
59$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 9 T + 93 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 11 T + 123 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + T + 111 T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 63 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 101 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 3 T + 147 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 10 T + 103 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 13 T + 197 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 153 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−12.96398085902552693312516278260, −12.78718315856322729734775489663, −12.10736179209522535271786520664, −11.75614757311944988792946563884, −10.85343415348044766005811362944, −10.32256216792915208114197744953, −9.622032481593677660586668674060, −9.540857999788061041666565340263, −8.816661258828038502276127464856, −8.641609712175748105035069728924, −7.939745056671385798664398401695, −7.30310767959864380296572794873, −6.48237622552898002112124554721, −6.10639120906328679343439521196, −4.97990707064660026254163588385, −4.93190557476243560191271324132, −3.78102691417589422800003718291, −3.77359646576267204480333738188, −2.49481129884255761010940407757, −1.67016810813577794666411149480, 1.67016810813577794666411149480, 2.49481129884255761010940407757, 3.77359646576267204480333738188, 3.78102691417589422800003718291, 4.93190557476243560191271324132, 4.97990707064660026254163588385, 6.10639120906328679343439521196, 6.48237622552898002112124554721, 7.30310767959864380296572794873, 7.939745056671385798664398401695, 8.641609712175748105035069728924, 8.816661258828038502276127464856, 9.540857999788061041666565340263, 9.622032481593677660586668674060, 10.32256216792915208114197744953, 10.85343415348044766005811362944, 11.75614757311944988792946563884, 12.10736179209522535271786520664, 12.78718315856322729734775489663, 12.96398085902552693312516278260

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