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Properties

Label 4-120e2-1.1-c1e2-0-4
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  − 2·3-s + 9-s + 4·13-s + 12·23-s + 25-s + 4·27-s + 4·37-s − 8·39-s − 12·47-s − 10·49-s + 24·59-s + 4·61-s − 24·69-s − 24·71-s + 4·73-s − 2·75-s − 11·81-s + 12·83-s + 4·97-s − 12·107-s + 4·109-s − 8·111-s + 4·117-s − 22·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 1.10·13-s + 2.50·23-s + 1/5·25-s + 0.769·27-s + 0.657·37-s − 1.28·39-s − 1.75·47-s − 1.42·49-s + 3.12·59-s + 0.512·61-s − 2.88·69-s − 2.84·71-s + 0.468·73-s − 0.230·75-s − 1.22·81-s + 1.31·83-s + 0.406·97-s − 1.16·107-s + 0.383·109-s − 0.759·111-s + 0.369·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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

\(L(1)\) \(\approx\) \(0.7675963187\)
\(L(\frac12)\) \(\approx\) \(0.7675963187\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.a_fa
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
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

−11.23781842834568778738111021684, −10.83473950301065015207539180976, −10.22754300302114228185381817663, −9.633261704500415779063377753491, −8.847850276232423443211930888771, −8.552217550781204179646223792184, −7.74818737522785607031159368119, −6.88656092970954199609923322632, −6.57891116465648258947670054106, −5.90138678053477774939394585997, −5.17606419420266207753384183426, −4.78130792717525308450176413839, −3.70610300459195515417889608627, −2.84705552122166969163585152299, −1.16915866227454488376692012454, 1.16915866227454488376692012454, 2.84705552122166969163585152299, 3.70610300459195515417889608627, 4.78130792717525308450176413839, 5.17606419420266207753384183426, 5.90138678053477774939394585997, 6.57891116465648258947670054106, 6.88656092970954199609923322632, 7.74818737522785607031159368119, 8.552217550781204179646223792184, 8.847850276232423443211930888771, 9.633261704500415779063377753491, 10.22754300302114228185381817663, 10.83473950301065015207539180976, 11.23781842834568778738111021684

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