Properties

Label 4-637e2-1.1-c1e2-0-8
Degree $4$
Conductor $405769$
Sign $1$
Analytic cond. $25.8721$
Root an. cond. $2.25532$
Motivic weight $1$
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  + 4·3-s + 6·9-s + 4·13-s − 4·16-s + 12·17-s − 6·23-s + 9·25-s − 4·27-s + 6·29-s + 16·39-s + 2·43-s − 16·48-s + 48·51-s − 18·53-s + 16·61-s − 24·69-s + 36·75-s − 18·79-s − 37·81-s + 24·87-s + 12·101-s − 24·107-s − 30·113-s + 24·117-s + 18·121-s + 127-s + 8·129-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s + 1.10·13-s − 16-s + 2.91·17-s − 1.25·23-s + 9/5·25-s − 0.769·27-s + 1.11·29-s + 2.56·39-s + 0.304·43-s − 2.30·48-s + 6.72·51-s − 2.47·53-s + 2.04·61-s − 2.88·69-s + 4.15·75-s − 2.02·79-s − 4.11·81-s + 2.57·87-s + 1.19·101-s − 2.32·107-s − 2.82·113-s + 2.21·117-s + 1.63·121-s + 0.0887·127-s + 0.704·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(405769\)    =    \(7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(25.8721\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 405769,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.610138610\)
\(L(\frac12)\) \(\approx\) \(4.610138610\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 153 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 113 T^{2} + p^{2} T^{4} \)
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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

−10.64234353303246160100060382416, −10.18803269324558568594717069969, −9.746101007605975765124687230574, −9.430037758233488900961057905407, −8.950676614316671573941948696336, −8.606407045003369079509021139899, −8.088788842302387893616119710109, −8.034339538881456639547981720508, −7.61738763057747751025397236972, −6.85266458704210460933678725820, −6.52707027617185373735316645633, −5.60119860321301535171771274622, −5.56138909409872972110325621622, −4.54523800628492560613720280741, −4.03358376185266144617444207717, −3.40767769512006132825402228318, −3.08957182611120525165091376776, −2.73064689323856064386024325161, −1.88813653210403727742605182787, −1.15690658711596644277879688543, 1.15690658711596644277879688543, 1.88813653210403727742605182787, 2.73064689323856064386024325161, 3.08957182611120525165091376776, 3.40767769512006132825402228318, 4.03358376185266144617444207717, 4.54523800628492560613720280741, 5.56138909409872972110325621622, 5.60119860321301535171771274622, 6.52707027617185373735316645633, 6.85266458704210460933678725820, 7.61738763057747751025397236972, 8.034339538881456639547981720508, 8.088788842302387893616119710109, 8.606407045003369079509021139899, 8.950676614316671573941948696336, 9.430037758233488900961057905407, 9.746101007605975765124687230574, 10.18803269324558568594717069969, 10.64234353303246160100060382416

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