Properties

Label 4-43904-1.1-c1e2-0-7
Degree $4$
Conductor $43904$
Sign $1$
Analytic cond. $2.79935$
Root an. cond. $1.29349$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s + 2·9-s − 14-s + 16-s + 2·18-s + 10·25-s − 28-s + 32-s + 2·36-s + 49-s + 10·50-s − 56-s − 2·63-s + 64-s + 2·72-s − 16·79-s − 5·81-s + 98-s + 10·100-s − 112-s − 12·113-s − 22·121-s − 2·126-s + 127-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 2/3·9-s − 0.267·14-s + 1/4·16-s + 0.471·18-s + 2·25-s − 0.188·28-s + 0.176·32-s + 1/3·36-s + 1/7·49-s + 1.41·50-s − 0.133·56-s − 0.251·63-s + 1/8·64-s + 0.235·72-s − 1.80·79-s − 5/9·81-s + 0.101·98-s + 100-s − 0.0944·112-s − 1.12·113-s − 2·121-s − 0.178·126-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(43904\)    =    \(2^{7} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(2.79935\)
Root analytic conductor: \(1.29349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{43904} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 43904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.105685636\)
\(L(\frac12)\) \(\approx\) \(2.105685636\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( 1 - T \)
7$C_1$ \( 1 + T \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−10.35429207920592299666236056630, −9.737571352224082577057668604337, −9.230234198592328844814622223277, −8.647620969069333470108951478696, −8.102096221801790937263883784465, −7.36010673208194959503095702760, −6.94438036583863366124517437556, −6.51987156391339999128357750064, −5.82258912862470940314930166189, −5.18598252263235902284418013548, −4.59160399106119589499226900700, −4.00728883804981885022921502277, −3.20128729290109899034222112750, −2.55522247581925182860702086473, −1.34090643042552658976895105787, 1.34090643042552658976895105787, 2.55522247581925182860702086473, 3.20128729290109899034222112750, 4.00728883804981885022921502277, 4.59160399106119589499226900700, 5.18598252263235902284418013548, 5.82258912862470940314930166189, 6.51987156391339999128357750064, 6.94438036583863366124517437556, 7.36010673208194959503095702760, 8.102096221801790937263883784465, 8.647620969069333470108951478696, 9.230234198592328844814622223277, 9.737571352224082577057668604337, 10.35429207920592299666236056630

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