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Properties

Label 4-2646e2-1.1-c1e2-0-10
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·5-s − 8-s + 2·10-s + 4·11-s + 2·13-s − 16-s − 7·19-s + 4·22-s − 6·23-s − 7·25-s + 2·26-s − 8·29-s + 4·31-s + 6·37-s − 7·38-s − 2·40-s + 12·41-s + 8·43-s − 6·46-s + 8·47-s − 7·50-s + 4·53-s + 8·55-s − 8·58-s − 4·59-s + 13·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.894·5-s − 0.353·8-s + 0.632·10-s + 1.20·11-s + 0.554·13-s − 1/4·16-s − 1.60·19-s + 0.852·22-s − 1.25·23-s − 7/5·25-s + 0.392·26-s − 1.48·29-s + 0.718·31-s + 0.986·37-s − 1.13·38-s − 0.316·40-s + 1.87·41-s + 1.21·43-s − 0.884·46-s + 1.16·47-s − 0.989·50-s + 0.549·53-s + 1.07·55-s − 1.05·58-s − 0.520·59-s + 1.66·61-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

\(L(1)\) \(\approx\) \(3.677906649\)
\(L(\frac12)\) \(\approx\) \(3.677906649\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.ac_aj
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.h_be
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.23.g_cd
29$C_2^2$ \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_bj
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_ab
41$C_2^2$ \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_dz
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ai_v
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_r
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_abl
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.an_ee
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_acl
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.71.ak_gl
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.73.o_et
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.79.al_bq
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_ed
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−9.152058993518840572734080700607, −8.742471107416447258100532916500, −8.249952523983594267131045080732, −7.931139018213358642795606502691, −7.60988523738923130747735231469, −6.88317493341133575334168231666, −6.71140557627281364441715370774, −6.12994127619579730224666716194, −5.97506015128172149149775059878, −5.56968975907132186214375588904, −5.47542544984589108479509040951, −4.44572916242978986060154518121, −4.18356286611476036372477265028, −3.93153099776452671846085613972, −3.80426351410436158100006812205, −2.66841255159820985244274110875, −2.52832423782121754325258172247, −1.89545495236671899988924430929, −1.44742321778561280540390847595, −0.55074313184025495314989677912, 0.55074313184025495314989677912, 1.44742321778561280540390847595, 1.89545495236671899988924430929, 2.52832423782121754325258172247, 2.66841255159820985244274110875, 3.80426351410436158100006812205, 3.93153099776452671846085613972, 4.18356286611476036372477265028, 4.44572916242978986060154518121, 5.47542544984589108479509040951, 5.56968975907132186214375588904, 5.97506015128172149149775059878, 6.12994127619579730224666716194, 6.71140557627281364441715370774, 6.88317493341133575334168231666, 7.60988523738923130747735231469, 7.931139018213358642795606502691, 8.249952523983594267131045080732, 8.742471107416447258100532916500, 9.152058993518840572734080700607

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