Properties

Label 4-39e4-1.1-c1e2-0-12
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $147.507$
Root an. cond. $3.48500$
Motivic weight $1$
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  − 4-s − 3·16-s − 12·17-s − 10·25-s − 12·29-s + 8·43-s − 2·49-s − 12·53-s − 4·61-s + 7·64-s + 12·68-s − 16·79-s + 10·100-s + 12·101-s − 16·103-s − 24·107-s + 12·113-s + 12·116-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1/2·4-s − 3/4·16-s − 2.91·17-s − 2·25-s − 2.22·29-s + 1.21·43-s − 2/7·49-s − 1.64·53-s − 0.512·61-s + 7/8·64-s + 1.45·68-s − 1.80·79-s + 100-s + 1.19·101-s − 1.57·103-s − 2.32·107-s + 1.12·113-s + 1.11·116-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(147.507\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2313441,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3 \( 1 \)
13 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−9.315039277666539569677827769091, −8.998764442342834100165097184971, −8.348056378507868857374601056029, −8.332257825284256696630759004412, −7.61922832362562448685817427313, −7.17482099423046659262209518890, −6.99861178288976733507776813503, −6.35312880114774201187140407245, −5.93944329937620013237831410201, −5.72983536965446822808509467013, −4.92051569999899627639582107947, −4.64589706219920012117607184766, −4.06264551255806293159961850586, −3.99881600976933982522072013808, −3.24276504929977198909387745568, −2.39455644807448585223645387954, −2.13982618907773264056735201748, −1.55463176831811738397772128768, 0, 0, 1.55463176831811738397772128768, 2.13982618907773264056735201748, 2.39455644807448585223645387954, 3.24276504929977198909387745568, 3.99881600976933982522072013808, 4.06264551255806293159961850586, 4.64589706219920012117607184766, 4.92051569999899627639582107947, 5.72983536965446822808509467013, 5.93944329937620013237831410201, 6.35312880114774201187140407245, 6.99861178288976733507776813503, 7.17482099423046659262209518890, 7.61922832362562448685817427313, 8.332257825284256696630759004412, 8.348056378507868857374601056029, 8.998764442342834100165097184971, 9.315039277666539569677827769091

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