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Properties

Label 4-2646e2-1.1-c1e2-0-25
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  + 2·2-s + 3·4-s + 2·5-s + 4·8-s + 4·10-s + 11-s + 6·13-s + 5·16-s − 5·17-s + 7·19-s + 6·20-s + 2·22-s + 4·23-s + 5·25-s + 12·26-s − 4·29-s − 12·31-s + 6·32-s − 10·34-s − 2·37-s + 14·38-s + 8·40-s + 3·41-s + 43-s + 3·44-s + 8·46-s + 10·50-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.41·8-s + 1.26·10-s + 0.301·11-s + 1.66·13-s + 5/4·16-s − 1.21·17-s + 1.60·19-s + 1.34·20-s + 0.426·22-s + 0.834·23-s + 25-s + 2.35·26-s − 0.742·29-s − 2.15·31-s + 1.06·32-s − 1.71·34-s − 0.328·37-s + 2.27·38-s + 1.26·40-s + 0.468·41-s + 0.152·43-s + 0.452·44-s + 1.17·46-s + 1.41·50-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

\(L(1)\) \(\approx\) \(10.62472919\)
\(L(\frac12)\) \(\approx\) \(10.62472919\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_ak
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2^2$ \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.17.f_i
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_an
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
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.ad_abg
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_dn
59$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.59.ao_gl
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.61.y_kg
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.67.aba_lr
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.b_acu
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.79.m_hm
83$C_2^2$ \( 1 - 16 T + 173 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.83.aq_gr
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.af_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

−8.917983940601465354939741345583, −8.894016847431953400980822889631, −8.266326861712310932284682929211, −7.80115269011844376530276373581, −7.18847173992713337705876950338, −7.13654956025678900262641677957, −6.48426227974975914469126083330, −6.46198629453036784790016244634, −5.74800514234666326904050439052, −5.58957686407212049532789372439, −5.09183853714789162143657185009, −5.02337073875204599186372971155, −4.10036243895398022835611411951, −3.87245379280636323075064588547, −3.54758530838991231870481696749, −3.02792937841832216522233550271, −2.45039758087918415575607234136, −1.99728705025649764464269649188, −1.43025321675836320593810492964, −0.885080629689844153505684757162, 0.885080629689844153505684757162, 1.43025321675836320593810492964, 1.99728705025649764464269649188, 2.45039758087918415575607234136, 3.02792937841832216522233550271, 3.54758530838991231870481696749, 3.87245379280636323075064588547, 4.10036243895398022835611411951, 5.02337073875204599186372971155, 5.09183853714789162143657185009, 5.58957686407212049532789372439, 5.74800514234666326904050439052, 6.46198629453036784790016244634, 6.48426227974975914469126083330, 7.13654956025678900262641677957, 7.18847173992713337705876950338, 7.80115269011844376530276373581, 8.266326861712310932284682929211, 8.894016847431953400980822889631, 8.917983940601465354939741345583

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