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Properties

Label 4-72e3-1.1-c1e2-0-12
Degree $4$
Conductor $373248$
Sign $1$
Analytic cond. $23.7986$
Root an. cond. $2.20870$
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  + 11-s + 11·17-s + 4·19-s − 3·25-s − 9·41-s + 17·43-s + 4·49-s − 5·59-s + 22·67-s − 4·73-s − 83-s − 27·89-s + 4·97-s − 9·107-s − 12·113-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  + 0.301·11-s + 2.66·17-s + 0.917·19-s − 3/5·25-s − 1.40·41-s + 2.59·43-s + 4/7·49-s − 0.650·59-s + 2.68·67-s − 0.468·73-s − 0.109·83-s − 2.86·89-s + 0.406·97-s − 0.870·107-s − 1.12·113-s − 1.72·121-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 + 2/13·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(373248\)    =    \(2^{9} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(23.7986\)
Root analytic conductor: \(2.20870\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 373248,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.141355156\)
\(L(\frac12)\) \(\approx\) \(2.141355156\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.11.ab_u
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.17.al_ck
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ae_bh
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.23.a_av
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.j_de
43$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.43.ar_fi
47$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.47.a_bf
53$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \) 2.53.a_j
59$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.f_es
61$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \) 2.61.a_aea
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.67.aw_jm
71$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.71.a_adm
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.e_ek
79$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.79.a_bc
83$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.83.b_ce
89$C_2$$\times$$C_2$ \( ( 1 + 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.bb_nu
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.97.ae_gg
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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

−8.591129551751807107548633168120, −8.150054461093459036082036858955, −7.81041985919391816762254860509, −7.32389799005898134326102442933, −6.99693484957480329940777751650, −6.29132945189820575521326718097, −5.68408943010533111082705292772, −5.49886907722190102660613456668, −5.01113874623755919933889698316, −4.11027902096070089773333604796, −3.77107099333903254350217640445, −3.13952470038265625264638456839, −2.60474986995835706880014704957, −1.56001359609216333213338763988, −0.907075890983809402213259270587, 0.907075890983809402213259270587, 1.56001359609216333213338763988, 2.60474986995835706880014704957, 3.13952470038265625264638456839, 3.77107099333903254350217640445, 4.11027902096070089773333604796, 5.01113874623755919933889698316, 5.49886907722190102660613456668, 5.68408943010533111082705292772, 6.29132945189820575521326718097, 6.99693484957480329940777751650, 7.32389799005898134326102442933, 7.81041985919391816762254860509, 8.150054461093459036082036858955, 8.591129551751807107548633168120

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