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Properties

Label 4-442368-1.1-c1e2-0-13
Degree $4$
Conductor $442368$
Sign $1$
Analytic cond. $28.2057$
Root an. cond. $2.30454$
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  − 3-s + 9-s + 4·11-s + 4·13-s + 4·23-s − 6·25-s − 27-s − 4·33-s − 4·39-s + 12·47-s + 6·49-s + 8·59-s − 24·61-s − 4·69-s + 20·71-s − 4·73-s + 6·75-s + 81-s + 4·83-s + 12·97-s + 4·99-s − 16·107-s + 12·109-s + 4·117-s − 6·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.20·11-s + 1.10·13-s + 0.834·23-s − 6/5·25-s − 0.192·27-s − 0.696·33-s − 0.640·39-s + 1.75·47-s + 6/7·49-s + 1.04·59-s − 3.07·61-s − 0.481·69-s + 2.37·71-s − 0.468·73-s + 0.692·75-s + 1/9·81-s + 0.439·83-s + 1.21·97-s + 0.402·99-s − 1.54·107-s + 1.14·109-s + 0.369·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(442368\)    =    \(2^{14} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(28.2057\)
Root analytic conductor: \(2.30454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 442368,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.782781879\)
\(L(\frac12)\) \(\approx\) \(1.782781879\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ae_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.17.a_g
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.ae_bu
29$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.29.a_as
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.41.a_g
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.43.a_acw
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.am_dq
53$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.53.a_aco
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ai_cs
61$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.y_kc
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.71.au_je
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.e_g
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.83.ae_gk
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.89.a_abi
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.am_ig
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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

−8.687467027461491245761968677984, −8.106375877486926970398856749633, −7.65197964585735512736691397962, −7.13724659280601977366598612007, −6.69782218459893280535642732383, −6.22345113435496137506090490864, −5.86395488338713502290327115202, −5.42469673324698400563075352090, −4.73589173484532398490404358679, −4.16447720115446200372378362101, −3.79366648248241696397598660767, −3.24054230197618884977375071097, −2.31884689878477876490769962103, −1.53216340101959582793506540836, −0.817139118401673994456221571538, 0.817139118401673994456221571538, 1.53216340101959582793506540836, 2.31884689878477876490769962103, 3.24054230197618884977375071097, 3.79366648248241696397598660767, 4.16447720115446200372378362101, 4.73589173484532398490404358679, 5.42469673324698400563075352090, 5.86395488338713502290327115202, 6.22345113435496137506090490864, 6.69782218459893280535642732383, 7.13724659280601977366598612007, 7.65197964585735512736691397962, 8.106375877486926970398856749633, 8.687467027461491245761968677984

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