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Properties

Label 4-5408e2-1.1-c1e2-0-12
Degree $4$
Conductor $29246464$
Sign $1$
Analytic cond. $1864.77$
Root an. cond. $6.57138$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 6·9-s − 2·17-s + 5·25-s + 10·29-s − 2·37-s + 10·41-s + 12·45-s − 14·49-s − 14·53-s + 10·61-s − 6·73-s + 27·81-s + 4·85-s − 20·89-s − 36·97-s + 2·101-s − 12·109-s + 14·113-s − 22·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s − 20·145-s + 149-s + ⋯
L(s)  = 1  − 0.894·5-s − 2·9-s − 0.485·17-s + 25-s + 1.85·29-s − 0.328·37-s + 1.56·41-s + 1.78·45-s − 2·49-s − 1.92·53-s + 1.28·61-s − 0.702·73-s + 3·81-s + 0.433·85-s − 2.11·89-s − 3.65·97-s + 0.199·101-s − 1.14·109-s + 1.31·113-s − 2·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.66·145-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29246464\)    =    \(2^{10} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1864.77\)
Root analytic conductor: \(6.57138\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 29246464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_ab
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_an
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.29.ak_ct
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2^2$ \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.41.ak_ch
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.53.o_fn
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_bn
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_abl
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.97.bk_ty
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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

−7.999793698076275763471941998069, −7.913821910496743656141746770164, −7.28023817101863750878745403939, −6.86272167886047903473130892528, −6.53222560342809395394481793333, −6.30691114145444500007409124254, −5.81965894239671974935774343134, −5.49916464042474488200018800200, −4.89333447972912507645196384162, −4.89128096766934071826383523754, −4.28509991184010126059378856235, −3.89076176106764088530380119409, −3.42834229640141922559355693504, −2.95758398968767848027872405391, −2.66119812688762528204277022342, −2.46166288495750056469208734819, −1.48341060479694159600545100678, −1.03050793454667933515781042821, 0, 0, 1.03050793454667933515781042821, 1.48341060479694159600545100678, 2.46166288495750056469208734819, 2.66119812688762528204277022342, 2.95758398968767848027872405391, 3.42834229640141922559355693504, 3.89076176106764088530380119409, 4.28509991184010126059378856235, 4.89128096766934071826383523754, 4.89333447972912507645196384162, 5.49916464042474488200018800200, 5.81965894239671974935774343134, 6.30691114145444500007409124254, 6.53222560342809395394481793333, 6.86272167886047903473130892528, 7.28023817101863750878745403939, 7.913821910496743656141746770164, 7.999793698076275763471941998069

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