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Properties

Label 4-442368-1.1-c1e2-0-12
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 + 10·17-s − 8·19-s − 2·25-s + 27-s − 4·33-s + 2·41-s − 4·43-s + 10·49-s + 10·51-s − 8·57-s − 4·59-s + 8·67-s + 20·73-s − 2·75-s + 81-s + 12·83-s + 18·89-s − 4·99-s − 4·107-s − 22·113-s + 6·121-s + 2·123-s + 127-s − 4·129-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s + 2.42·17-s − 1.83·19-s − 2/5·25-s + 0.192·27-s − 0.696·33-s + 0.312·41-s − 0.609·43-s + 10/7·49-s + 1.40·51-s − 1.05·57-s − 0.520·59-s + 0.977·67-s + 2.34·73-s − 0.230·75-s + 1/9·81-s + 1.31·83-s + 1.90·89-s − 0.402·99-s − 0.386·107-s − 2.06·113-s + 6/11·121-s + 0.180·123-s + 0.0887·127-s − 0.352·129-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

\(L(1)\) \(\approx\) \(2.080560907\)
\(L(\frac12)\) \(\approx\) \(2.080560907\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.e_k
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.17.ak_cg
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ac_de
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.e_cc
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.47.a_c
53$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.53.a_ack
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.e_di
61$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.61.a_bm
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ai_di
71$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.71.a_adq
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.73.au_iw
79$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.79.a_dy
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.as_gw
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−8.328393326310747805121223539889, −8.183981963669871922709624852076, −7.76746773465333335664011252853, −7.44056605684254858936792029527, −6.74709799243706617042890539415, −6.31879596862550091228684328087, −5.69903289863592517141596692118, −5.30790916053724469270804254899, −4.84335964197004779748227925945, −4.09540802495458556946701509560, −3.63776461821432417761955516907, −3.08552310305503150051979769154, −2.40895468931649609205020815967, −1.88651998003305839287872142530, −0.75540255720867727176094743356, 0.75540255720867727176094743356, 1.88651998003305839287872142530, 2.40895468931649609205020815967, 3.08552310305503150051979769154, 3.63776461821432417761955516907, 4.09540802495458556946701509560, 4.84335964197004779748227925945, 5.30790916053724469270804254899, 5.69903289863592517141596692118, 6.31879596862550091228684328087, 6.74709799243706617042890539415, 7.44056605684254858936792029527, 7.76746773465333335664011252853, 8.183981963669871922709624852076, 8.328393326310747805121223539889

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