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Properties

Label 4-1379e2-1.1-c1e2-0-0
Degree $4$
Conductor $1901641$
Sign $-1$
Analytic cond. $121.250$
Root an. cond. $3.31833$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s − 3·7-s − 8·8-s − 6·9-s + 8·11-s + 12·14-s − 4·16-s + 24·18-s − 32·22-s − 6·23-s − 10·25-s − 24·28-s + 14·29-s + 32·32-s − 48·36-s + 14·37-s + 2·43-s + 64·44-s + 24·46-s + 2·49-s + 40·50-s + 20·53-s + 24·56-s − 56·58-s + 18·63-s − 64·64-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s − 1.13·7-s − 2.82·8-s − 2·9-s + 2.41·11-s + 3.20·14-s − 16-s + 5.65·18-s − 6.82·22-s − 1.25·23-s − 2·25-s − 4.53·28-s + 2.59·29-s + 5.65·32-s − 8·36-s + 2.30·37-s + 0.304·43-s + 9.64·44-s + 3.53·46-s + 2/7·49-s + 5.65·50-s + 2.74·53-s + 3.20·56-s − 7.35·58-s + 2.26·63-s − 8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1901641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1901641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1901641\)    =    \(7^{2} \cdot 197^{2}\)
Sign: $-1$
Analytic conductor: \(121.250\)
Root analytic conductor: \(3.31833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1901641,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad7$C_2$ \( 1 + 3 T + p T^{2} \)
197$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−7.87206983988291315577291885956, −7.31760208234592901601947662201, −6.92472041340395206972505718477, −6.27518514436818440070124891939, −6.17591940844279569664203110179, −6.02545888580051406290319703687, −5.01661231949680571647317933664, −4.15785519193072071590213469433, −4.05398686955634354061385342731, −3.28092630625484404757155593416, −2.40495253382643376982803890844, −2.33277116067346011831864956690, −1.25480551044621632882557576276, −0.77820321769958627649568491953, 0, 0.77820321769958627649568491953, 1.25480551044621632882557576276, 2.33277116067346011831864956690, 2.40495253382643376982803890844, 3.28092630625484404757155593416, 4.05398686955634354061385342731, 4.15785519193072071590213469433, 5.01661231949680571647317933664, 6.02545888580051406290319703687, 6.17591940844279569664203110179, 6.27518514436818440070124891939, 6.92472041340395206972505718477, 7.31760208234592901601947662201, 7.87206983988291315577291885956

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