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Properties

Label 4-39e2-1.1-c16e2-0-0
Degree $4$
Conductor $1521$
Sign $1$
Analytic cond. $4007.72$
Root an. cond. $7.95654$
Motivic weight $16$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.31e4·3-s − 8.32e4·4-s + 1.29e8·9-s + 1.09e9·12-s − 1.63e9·13-s + 2.64e9·16-s − 2.89e11·25-s − 1.12e12·27-s − 1.07e13·36-s + 2.14e13·39-s + 1.14e13·43-s − 3.46e13·48-s + 6.64e13·49-s + 1.35e14·52-s − 1.34e14·61-s + 1.37e14·64-s + 3.80e15·75-s + 5.12e15·79-s + 9.26e15·81-s + 2.41e16·100-s − 1.92e16·103-s + 9.41e16·108-s − 2.10e17·117-s − 5.42e16·121-s + 127-s − 1.49e17·129-s + 131-s + ⋯
L(s)  = 1  − 2·3-s − 1.27·4-s + 3·9-s + 2.54·12-s − 2·13-s + 0.615·16-s − 1.89·25-s − 4·27-s − 3.81·36-s + 4·39-s + 0.976·43-s − 1.23·48-s + 2·49-s + 2.54·52-s − 0.702·61-s + 0.488·64-s + 3.79·75-s + 3.37·79-s + 5·81-s + 2.41·100-s − 1.52·103-s + 5.08·108-s − 6·117-s − 1.17·121-s − 1.95·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+8)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(4007.72\)
Root analytic conductor: \(7.95654\)
Motivic weight: \(16\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1521,\ (\ :8, 8),\ 1)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.5805407834\)
\(L(\frac12)\) \(\approx\) \(0.5805407834\)
\(L(9)\) 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$C_1$ \( ( 1 + p^{8} T )^{2} \)
13$C_1$ \( ( 1 + p^{8} T )^{2} \)
good2$C_2^2$ \( 1 + 83297 T^{2} + p^{32} T^{4} \)
5$C_2^2$ \( 1 + 289675461506 T^{2} + p^{32} T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
11$C_2^2$ \( 1 + 54202329825294722 T^{2} + p^{32} T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
41$C_2^2$ \( 1 - \)\(73\!\cdots\!18\)\( T^{2} + p^{32} T^{4} \)
43$C_2$ \( ( 1 - 5705869713602 T + p^{16} T^{2} )^{2} \)
47$C_2^2$ \( 1 + \)\(43\!\cdots\!42\)\( T^{2} + p^{32} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
59$C_2^2$ \( 1 + \)\(24\!\cdots\!82\)\( T^{2} + p^{32} T^{4} \)
61$C_2$ \( ( 1 + 67294792292162 T + p^{16} T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
71$C_2^2$ \( 1 - \)\(83\!\cdots\!58\)\( T^{2} + p^{32} T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
79$C_2$ \( ( 1 - 2561582976026878 T + p^{16} T^{2} )^{2} \)
83$C_2^2$ \( 1 + \)\(79\!\cdots\!62\)\( T^{2} + p^{32} T^{4} \)
89$C_2^2$ \( 1 + \)\(13\!\cdots\!22\)\( T^{2} + p^{32} T^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
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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

−12.74576118383150942293209197419, −12.44307563027334657676184870186, −11.90801557759617427259371157544, −11.45208472596449195463569632112, −10.50530540297353424826130709093, −10.21852649759581016795486103166, −9.385762584852896348374650936635, −9.323585298419995847425908253923, −7.85036639794526037405599366718, −7.51895770115127298416409780230, −6.72347386103220989747054506375, −5.99715213428062174498945372507, −5.19256748763859779589517399768, −5.12602799999166937472686289088, −4.14761280302265461794975315748, −4.00635816902142681546238053008, −2.44100212569648923144415060482, −1.66207611239276712399570836822, −0.54059792851842691528389827166, −0.44770781067686634885897860705, 0.44770781067686634885897860705, 0.54059792851842691528389827166, 1.66207611239276712399570836822, 2.44100212569648923144415060482, 4.00635816902142681546238053008, 4.14761280302265461794975315748, 5.12602799999166937472686289088, 5.19256748763859779589517399768, 5.99715213428062174498945372507, 6.72347386103220989747054506375, 7.51895770115127298416409780230, 7.85036639794526037405599366718, 9.323585298419995847425908253923, 9.385762584852896348374650936635, 10.21852649759581016795486103166, 10.50530540297353424826130709093, 11.45208472596449195463569632112, 11.90801557759617427259371157544, 12.44307563027334657676184870186, 12.74576118383150942293209197419

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