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Properties

Label 4-43904-1.1-c1e2-0-9
Degree $4$
Conductor $43904$
Sign $1$
Analytic cond. $2.79935$
Root an. cond. $1.29349$
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-s + 4-s − 7-s + 8-s + 5·9-s − 14-s + 16-s + 3·17-s + 5·18-s − 3·23-s − 2·25-s − 28-s − 6·31-s + 32-s + 3·34-s + 5·36-s + 9·41-s − 3·46-s − 15·47-s + 49-s − 2·50-s − 56-s − 6·62-s − 5·63-s + 64-s + 3·68-s + 15·71-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 5/3·9-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 1.17·18-s − 0.625·23-s − 2/5·25-s − 0.188·28-s − 1.07·31-s + 0.176·32-s + 0.514·34-s + 5/6·36-s + 1.40·41-s − 0.442·46-s − 2.18·47-s + 1/7·49-s − 0.282·50-s − 0.133·56-s − 0.762·62-s − 0.629·63-s + 1/8·64-s + 0.363·68-s + 1.78·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(43904\)    =    \(2^{7} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(2.79935\)
Root analytic conductor: \(1.29349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 43904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.213605989\)
\(L(\frac12)\) \(\approx\) \(2.213605989\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_1$ \( 1 + T \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.13.a_u
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.ad_q
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.d_bc
29$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \) 2.29.a_br
31$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.g_cp
37$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.37.a_af
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.aj_de
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.43.a_bi
47$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.p_fa
53$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.53.a_dh
59$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.59.a_x
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.61.a_abo
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.67.a_bi
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.71.ap_ho
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.g_ec
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.79.al_gm
83$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.83.a_f
89$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.p_ig
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.m_gk
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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

−10.05678375381477042318874611224, −9.756948225725246098638511465202, −9.502508101509796184785562727000, −8.572471043036661544792716627202, −7.959398509703870639269961294849, −7.40680873284187474787785819703, −7.08975345681600839738665962505, −6.35386202430917741466320291891, −5.93928509818591661972668925621, −5.13656851188741939169424055778, −4.60634381509827538369531600859, −3.84629395034157933487320961776, −3.48844498554738898579609102006, −2.34612635281934047519280732349, −1.42882639006212863066694207280, 1.42882639006212863066694207280, 2.34612635281934047519280732349, 3.48844498554738898579609102006, 3.84629395034157933487320961776, 4.60634381509827538369531600859, 5.13656851188741939169424055778, 5.93928509818591661972668925621, 6.35386202430917741466320291891, 7.08975345681600839738665962505, 7.40680873284187474787785819703, 7.959398509703870639269961294849, 8.572471043036661544792716627202, 9.502508101509796184785562727000, 9.756948225725246098638511465202, 10.05678375381477042318874611224

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