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Properties

Label 4-34e4-1.1-c0e2-0-3
Degree $4$
Conductor $1336336$
Sign $1$
Analytic cond. $0.332835$
Root an. cond. $0.759551$
Motivic weight $0$
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  + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 5·16-s − 4·18-s + 6·32-s − 6·36-s − 2·49-s + 7·64-s − 8·72-s + 3·81-s − 4·98-s − 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 6·162-s + 163-s + 167-s − 2·169-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 5·16-s − 4·18-s + 6·32-s − 6·36-s − 2·49-s + 7·64-s − 8·72-s + 3·81-s − 4·98-s − 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 6·162-s + 163-s + 167-s − 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1336336\)    =    \(2^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(0.332835\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1336336,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.410826383\)
\(L(\frac12)\) \(\approx\) \(3.410826383\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
17 \( 1 \)
good3$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−10.37782133896535897058727791420, −9.959446504008254711568932284596, −9.473336792913580224820822284835, −8.847343217806489063264158348341, −8.372049316369935133831786348577, −8.123104844868391067328305122397, −7.46786936584930805246552501014, −7.32600854090312147157857780631, −6.44387746314973525475619203676, −6.28213338680234465645477220488, −6.06216260690761447288166444172, −5.37658945327439208851325846734, −4.98648292657619979258461325750, −4.91609366761207145121749398379, −3.86530813933462331101078733180, −3.79083432070679022376672819701, −2.98841318741524794401284343643, −2.78962838836108292014508091453, −2.20994499824788278960940488802, −1.41958524975628106160777037168, 1.41958524975628106160777037168, 2.20994499824788278960940488802, 2.78962838836108292014508091453, 2.98841318741524794401284343643, 3.79083432070679022376672819701, 3.86530813933462331101078733180, 4.91609366761207145121749398379, 4.98648292657619979258461325750, 5.37658945327439208851325846734, 6.06216260690761447288166444172, 6.28213338680234465645477220488, 6.44387746314973525475619203676, 7.32600854090312147157857780631, 7.46786936584930805246552501014, 8.123104844868391067328305122397, 8.372049316369935133831786348577, 8.847343217806489063264158348341, 9.473336792913580224820822284835, 9.959446504008254711568932284596, 10.37782133896535897058727791420

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