Properties

Label 4-281216-1.1-c1e2-0-7
Degree $4$
Conductor $281216$
Sign $-1$
Analytic cond. $17.9305$
Root an. cond. $2.05777$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 6·5-s + 8-s − 5·9-s + 6·10-s − 12·11-s − 13-s + 16-s − 6·17-s − 5·18-s − 4·19-s + 6·20-s − 12·22-s + 17·25-s − 26-s + 32-s − 6·34-s − 5·36-s + 14·37-s − 4·38-s + 6·40-s − 12·44-s − 30·45-s − 13·49-s + 17·50-s − 52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 2.68·5-s + 0.353·8-s − 5/3·9-s + 1.89·10-s − 3.61·11-s − 0.277·13-s + 1/4·16-s − 1.45·17-s − 1.17·18-s − 0.917·19-s + 1.34·20-s − 2.55·22-s + 17/5·25-s − 0.196·26-s + 0.176·32-s − 1.02·34-s − 5/6·36-s + 2.30·37-s − 0.648·38-s + 0.948·40-s − 1.80·44-s − 4.47·45-s − 1.85·49-s + 2.40·50-s − 0.138·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 281216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(281216\)    =    \(2^{7} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(17.9305\)
Root analytic conductor: \(2.05777\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 281216,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_1$ \( 1 - T \)
13$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.631851122063147597031259612932, −8.123374860892634538751745807462, −7.85100333312502477176899037562, −7.06762116885878584702639951011, −6.43741307804194710670707219872, −5.95269101287348431555436217017, −5.62758036787742512501124982836, −5.56442463767257268744193982616, −4.68671862775252250000074194944, −4.66492539504114224399994882224, −2.92515547495828704252974886323, −2.86630826934752796258245605486, −2.19431989234797768682378556675, −2.10253053611975333597513041784, 0, 2.10253053611975333597513041784, 2.19431989234797768682378556675, 2.86630826934752796258245605486, 2.92515547495828704252974886323, 4.66492539504114224399994882224, 4.68671862775252250000074194944, 5.56442463767257268744193982616, 5.62758036787742512501124982836, 5.95269101287348431555436217017, 6.43741307804194710670707219872, 7.06762116885878584702639951011, 7.85100333312502477176899037562, 8.123374860892634538751745807462, 8.631851122063147597031259612932

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