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Properties

Label 4-2646e2-1.1-c1e2-0-2
Degree $4$
Conductor $7001316$
Sign $1$
Analytic cond. $446.409$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4·5-s + 8-s − 4·10-s − 2·11-s − 6·13-s − 16-s + 5·17-s − 7·19-s + 2·22-s − 8·23-s + 2·25-s + 6·26-s − 4·29-s − 6·31-s − 5·34-s − 2·37-s + 7·38-s + 4·40-s − 3·41-s + 43-s + 8·46-s − 2·50-s + 12·53-s − 8·55-s + 4·58-s + 7·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.78·5-s + 0.353·8-s − 1.26·10-s − 0.603·11-s − 1.66·13-s − 1/4·16-s + 1.21·17-s − 1.60·19-s + 0.426·22-s − 1.66·23-s + 2/5·25-s + 1.17·26-s − 0.742·29-s − 1.07·31-s − 0.857·34-s − 0.328·37-s + 1.13·38-s + 0.632·40-s − 0.468·41-s + 0.152·43-s + 1.17·46-s − 0.282·50-s + 1.64·53-s − 1.07·55-s + 0.525·58-s + 0.911·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7001316\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(446.409\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7001316,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4089761858\)
\(L(\frac12)\) \(\approx\) \(0.4089761858\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.11.c_x
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2^2$ \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.17.af_i
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.h_be
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_an
31$C_2^2$ \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_f
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_dn
59$C_2^2$ \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.59.ah_ak
61$C_2^2$ \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_df
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.67.n_dy
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) 2.73.ab_acu
79$C_2^2$ \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_abr
83$C_2^2$ \( 1 + 16 T + 173 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.83.q_gr
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.f_acu
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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

−9.217637976670619819793185243771, −8.810661984264330921149051843940, −8.168340322679005512462088057496, −7.937426045493439712861710692592, −7.76311535284028798622245006652, −7.13271091214475350525134201749, −6.80253076925883842234073445890, −6.43617715905182165911512521095, −5.68649778031052664928837546195, −5.66489699492468480610430555085, −5.43887385743767058580437677234, −4.88188944588736879478920717869, −4.11382886085275281724842610614, −4.09074320107573478716896028619, −3.25544249892602445848897696443, −2.55245853285665093412408415442, −2.23955877835577022395845030529, −1.88657746326511693864336009756, −1.43636455557231033643852398841, −0.21945234658843250227167295579, 0.21945234658843250227167295579, 1.43636455557231033643852398841, 1.88657746326511693864336009756, 2.23955877835577022395845030529, 2.55245853285665093412408415442, 3.25544249892602445848897696443, 4.09074320107573478716896028619, 4.11382886085275281724842610614, 4.88188944588736879478920717869, 5.43887385743767058580437677234, 5.66489699492468480610430555085, 5.68649778031052664928837546195, 6.43617715905182165911512521095, 6.80253076925883842234073445890, 7.13271091214475350525134201749, 7.76311535284028798622245006652, 7.937426045493439712861710692592, 8.168340322679005512462088057496, 8.810661984264330921149051843940, 9.217637976670619819793185243771

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