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Properties

Label 4-120e2-1.1-c1e2-0-5
Degree $4$
Conductor $14400$
Sign $1$
Analytic cond. $0.918156$
Root an. cond. $0.978879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·8-s + 9-s − 16-s + 4·17-s − 18-s + 25-s − 5·32-s − 4·34-s − 36-s + 20·41-s + 16·47-s − 14·49-s − 50-s + 7·64-s − 4·68-s − 16·71-s + 3·72-s + 20·73-s + 81-s − 20·82-s − 12·89-s − 16·94-s + 4·97-s + 14·98-s − 100-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s + 1/3·9-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 1/5·25-s − 0.883·32-s − 0.685·34-s − 1/6·36-s + 3.12·41-s + 2.33·47-s − 2·49-s − 0.141·50-s + 7/8·64-s − 0.485·68-s − 1.89·71-s + 0.353·72-s + 2.34·73-s + 1/9·81-s − 2.20·82-s − 1.27·89-s − 1.65·94-s + 0.406·97-s + 1.41·98-s − 0.0999·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.918156\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6935617555\)
\(L(\frac12)\) \(\approx\) \(0.6935617555\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−10.87392441792526097751028478771, −10.67892245123374144028858824682, −9.907104671360377962786893632766, −9.523451675812284265792380668213, −9.080341213166513123570518780286, −8.438902886313495638742074033009, −7.66488013441745380243230842523, −7.60125180356652570547951985370, −6.67875901406960725991694880905, −5.85130665991850657402735978605, −5.23920392624592057055772361749, −4.38871739502786245271092399731, −3.82349173075266898382654912582, −2.63252540175087701435373844808, −1.18815390030287753587945796282, 1.18815390030287753587945796282, 2.63252540175087701435373844808, 3.82349173075266898382654912582, 4.38871739502786245271092399731, 5.23920392624592057055772361749, 5.85130665991850657402735978605, 6.67875901406960725991694880905, 7.60125180356652570547951985370, 7.66488013441745380243230842523, 8.438902886313495638742074033009, 9.080341213166513123570518780286, 9.523451675812284265792380668213, 9.907104671360377962786893632766, 10.67892245123374144028858824682, 10.87392441792526097751028478771

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