Properties

Label 4-1216e2-1.1-c3e2-0-4
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $5147.53$
Root an. cond. $8.47032$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·5-s − 44·7-s − 51·9-s + 40·11-s + 20·13-s − 8·15-s + 46·17-s + 38·19-s + 88·21-s + 220·23-s + 134·25-s + 158·27-s + 84·29-s + 84·31-s − 80·33-s − 176·35-s − 304·37-s − 40·39-s + 168·41-s − 372·43-s − 204·45-s − 584·47-s + 859·49-s − 92·51-s − 484·53-s + 160·55-s + ⋯
L(s)  = 1  − 0.384·3-s + 0.357·5-s − 2.37·7-s − 1.88·9-s + 1.09·11-s + 0.426·13-s − 0.137·15-s + 0.656·17-s + 0.458·19-s + 0.914·21-s + 1.99·23-s + 1.07·25-s + 1.12·27-s + 0.537·29-s + 0.486·31-s − 0.422·33-s − 0.849·35-s − 1.35·37-s − 0.164·39-s + 0.639·41-s − 1.31·43-s − 0.675·45-s − 1.81·47-s + 2.50·49-s − 0.252·51-s − 1.25·53-s + 0.392·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5147.53\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1478656,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
19$C_1$ \( ( 1 - p T )^{2} \)
good3$C_2$ \( ( 1 + T + p^{3} T^{2} )^{2} \)
5$D_{4}$ \( 1 - 4 T - 118 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 44 T + 1077 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 40 T + 2690 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 20 T + 3657 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 46 T + 259 p T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 220 T + 1483 p T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 84 T + 16969 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 84 T + 31214 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 304 T + 121062 T^{2} + 304 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 168 T + 107698 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 372 T + 193238 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 584 T + 239342 T^{2} + 584 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 484 T + 255041 T^{2} + 484 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 1074 T + 697639 T^{2} + 1074 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 88 T + 437670 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 1430 T + 1039839 T^{2} - 1430 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1848 T + 1569226 T^{2} + 1848 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1294 T + 1190691 T^{2} + 1294 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 832 T + 593322 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 528 T + 459970 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1844 T + 1750754 T^{2} - 1844 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 364 T + 1606998 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.128769234310573620005228056931, −8.917893205469915773158969755520, −8.485242741495349942554915888680, −8.083738722835255727420789698462, −7.33711514859523866606934098749, −6.75128826479355224509431381338, −6.56410156099193031788154077670, −6.41848853858587837272007550388, −5.79728242906429228223159347915, −5.55413727175344320027893379523, −4.93072113533786590667549632860, −4.55136195158787050176654012662, −3.49115604823327394888079237802, −3.28167449050424977321876635221, −3.02263423017133149104774076181, −2.71232307156100525535306455302, −1.45031275962286229185646455701, −1.04993588526087211607630818539, 0, 0, 1.04993588526087211607630818539, 1.45031275962286229185646455701, 2.71232307156100525535306455302, 3.02263423017133149104774076181, 3.28167449050424977321876635221, 3.49115604823327394888079237802, 4.55136195158787050176654012662, 4.93072113533786590667549632860, 5.55413727175344320027893379523, 5.79728242906429228223159347915, 6.41848853858587837272007550388, 6.56410156099193031788154077670, 6.75128826479355224509431381338, 7.33711514859523866606934098749, 8.083738722835255727420789698462, 8.485242741495349942554915888680, 8.917893205469915773158969755520, 9.128769234310573620005228056931

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