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Properties

Label 4-180e2-1.1-c1e2-0-0
Degree $4$
Conductor $32400$
Sign $1$
Analytic cond. $2.06585$
Root an. cond. $1.19887$
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  − 2·4-s + 4·13-s + 4·16-s + 25-s + 4·37-s + 8·49-s − 8·52-s + 16·61-s − 8·64-s + 28·73-s − 20·97-s − 2·100-s − 8·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 4-s + 1.10·13-s + 16-s + 1/5·25-s + 0.657·37-s + 8/7·49-s − 1.10·52-s + 2.04·61-s − 64-s + 3.27·73-s − 2.03·97-s − 1/5·100-s − 0.766·109-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(32400\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2.06585\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 32400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.038842612\)
\(L(\frac12)\) \(\approx\) \(1.038842612\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
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$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.23.a_bo
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.31.a_bi
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$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.43.a_adc
47$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \) 2.47.a_ace
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.59.a_w
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.67.a_adc
71$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \) 2.71.a_eo
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \) 2.79.a_afe
83$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.83.a_ei
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−10.42622977065269933855893812010, −9.813157066547424337510185330525, −9.429118435037894227504984756223, −8.878904432761376537547579456374, −8.288828315301589688901603419147, −8.091937983271484055936897365069, −7.23363581552713227021582441250, −6.61668149635008544378180724561, −5.94171836444501009690226869744, −5.39504256604270189333590479848, −4.77867221017145857963863619851, −3.93826359666462119089513224541, −3.60167852700477778234866407088, −2.46590351451006153706317864020, −1.05279196353716811796825860551, 1.05279196353716811796825860551, 2.46590351451006153706317864020, 3.60167852700477778234866407088, 3.93826359666462119089513224541, 4.77867221017145857963863619851, 5.39504256604270189333590479848, 5.94171836444501009690226869744, 6.61668149635008544378180724561, 7.23363581552713227021582441250, 8.091937983271484055936897365069, 8.288828315301589688901603419147, 8.878904432761376537547579456374, 9.429118435037894227504984756223, 9.813157066547424337510185330525, 10.42622977065269933855893812010

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