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Properties

Label 4-864e2-1.1-c1e2-0-30
Degree $4$
Conductor $746496$
Sign $-1$
Analytic cond. $47.5972$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·11-s + 3·17-s + 2·19-s − 25-s + 9·41-s − 7·43-s − 10·49-s − 15·59-s − 10·67-s − 14·73-s + 21·83-s + 3·89-s − 8·97-s − 33·107-s + 12·113-s − 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + ⋯
L(s)  = 1  − 0.904·11-s + 0.727·17-s + 0.458·19-s − 1/5·25-s + 1.40·41-s − 1.06·43-s − 1.42·49-s − 1.95·59-s − 1.22·67-s − 1.63·73-s + 2.30·83-s + 0.317·89-s − 0.812·97-s − 3.19·107-s + 1.12·113-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.307·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(746496\)    =    \(2^{10} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(47.5972\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 746496,\ (\ :1/2, 1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.d_w
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.13.a_e
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.ad_q
19$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.ac_d
23$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.23.a_t
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.37.a_acq
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.aj_bu
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.h_da
47$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.47.a_bf
53$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.53.a_abd
59$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.p_gq
61$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.61.a_ao
67$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.k_fu
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.71.a_cs
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.o_he
79$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.79.a_aeg
83$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.83.av_im
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.ad_ge
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.i_gs
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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

−8.045341867674414842050062095173, −7.66180435710951130569001736977, −7.29421034537608693324627633554, −6.76000922076055942065765408833, −6.09746805300394433377005791914, −5.91390690868343933022328821893, −5.25767895579495833664005029602, −4.87948214229771902976184113537, −4.40081430962366242507071821991, −3.71641967585737752464275477547, −3.10131259833778906742446300429, −2.78538657920062050817305967176, −1.91508024921452348895485944175, −1.21145250786458108063321446106, 0, 1.21145250786458108063321446106, 1.91508024921452348895485944175, 2.78538657920062050817305967176, 3.10131259833778906742446300429, 3.71641967585737752464275477547, 4.40081430962366242507071821991, 4.87948214229771902976184113537, 5.25767895579495833664005029602, 5.91390690868343933022328821893, 6.09746805300394433377005791914, 6.76000922076055942065765408833, 7.29421034537608693324627633554, 7.66180435710951130569001736977, 8.045341867674414842050062095173

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