Properties

Label 4-666e2-1.1-c1e2-0-10
Degree $4$
Conductor $443556$
Sign $1$
Analytic cond. $28.2815$
Root an. cond. $2.30608$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 5-s + 8-s + 10-s − 4·11-s + 6·13-s − 16-s + 3·17-s − 2·19-s + 4·22-s + 12·23-s + 5·25-s − 6·26-s + 18·29-s + 20·31-s − 3·34-s − 37-s + 2·38-s − 40-s + 3·41-s − 16·43-s − 12·46-s − 4·47-s + 7·49-s − 5·50-s − 6·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.447·5-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.66·13-s − 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.852·22-s + 2.50·23-s + 25-s − 1.17·26-s + 3.34·29-s + 3.59·31-s − 0.514·34-s − 0.164·37-s + 0.324·38-s − 0.158·40-s + 0.468·41-s − 2.43·43-s − 1.76·46-s − 0.583·47-s + 49-s − 0.707·50-s − 0.824·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(443556\)    =    \(2^{2} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(28.2815\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 443556,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.423899452\)
\(L(\frac12)\) \(\approx\) \(1.423899452\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
37$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.29.as_fj
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.47.e_du
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_f
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.61.af_abk
67$C_2^2$ \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_abf
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.71.a_act
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_abr
83$C_2^2$ \( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_adb
89$C_2^2$ \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.89.n_dc
97$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.97.ag_hv
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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

−10.56182993229973597377111061432, −10.43137175751561101728152793698, −9.842415721938944643242407657148, −9.620051478534108353908660449138, −8.605848493459808768427459796294, −8.517268320953162470790712985072, −8.250448953941239666876287998488, −8.142501523555363276404054290578, −7.05122773389641291060436914852, −6.90405443356493407183197926199, −6.44391746946068000561569181949, −5.92060136837988213742049818426, −5.03788140805041894714386836674, −4.83800202762840085182671713931, −4.43072838136009752251176800975, −3.48482059807355046542627915642, −2.82793741257533133459222477835, −2.76348563949483340092135235285, −1.05977880508536725700771349528, −1.01936612968966133430867180724, 1.01936612968966133430867180724, 1.05977880508536725700771349528, 2.76348563949483340092135235285, 2.82793741257533133459222477835, 3.48482059807355046542627915642, 4.43072838136009752251176800975, 4.83800202762840085182671713931, 5.03788140805041894714386836674, 5.92060136837988213742049818426, 6.44391746946068000561569181949, 6.90405443356493407183197926199, 7.05122773389641291060436914852, 8.142501523555363276404054290578, 8.250448953941239666876287998488, 8.517268320953162470790712985072, 8.605848493459808768427459796294, 9.620051478534108353908660449138, 9.842415721938944643242407657148, 10.43137175751561101728152793698, 10.56182993229973597377111061432

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