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Properties

Label 4-1456e2-1.1-c1e2-0-3
Degree $4$
Conductor $2119936$
Sign $1$
Analytic cond. $135.168$
Root an. cond. $3.40972$
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  + 3·3-s − 6·5-s + 7-s + 3·9-s + 4·11-s − 5·13-s − 18·15-s − 2·17-s − 4·19-s + 3·21-s − 23-s + 17·25-s + 2·29-s − 20·31-s + 12·33-s − 6·35-s − 4·37-s − 15·39-s − 12·41-s − 18·45-s + 20·47-s − 6·51-s − 8·53-s − 24·55-s − 12·57-s + 9·59-s − 13·61-s + ⋯
L(s)  = 1  + 1.73·3-s − 2.68·5-s + 0.377·7-s + 9-s + 1.20·11-s − 1.38·13-s − 4.64·15-s − 0.485·17-s − 0.917·19-s + 0.654·21-s − 0.208·23-s + 17/5·25-s + 0.371·29-s − 3.59·31-s + 2.08·33-s − 1.01·35-s − 0.657·37-s − 2.40·39-s − 1.87·41-s − 2.68·45-s + 2.91·47-s − 0.840·51-s − 1.09·53-s − 3.23·55-s − 1.58·57-s + 1.17·59-s − 1.66·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2119936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2119936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2119936\)    =    \(2^{8} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(135.168\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2119936,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6210705780\)
\(L(\frac12)\) \(\approx\) \(0.6210705780\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - T + T^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) 2.3.ad_g
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_f
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_an
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_ad
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) 2.23.b_aw
29$C_2^2$ \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_az
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.31.u_gg
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_av
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dz
43$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.43.a_abr
47$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.47.au_hm
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.53.i_es
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_w
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.n_ee
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_acl
71$C_2^2$ \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.71.p_fy
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.83.aq_iw
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

−10.13213804910786260290176379548, −9.053432103286485331822425981631, −8.794012175979894117326943356371, −8.580426624740774002663258894686, −8.196739503927126331638371543454, −7.63935688462243273270055874365, −7.45016227287831454229440498385, −7.17333273231704129238110152830, −6.91907875715317599664719444575, −6.09256400202071666107587540515, −5.52315674764975307326848703611, −4.68532767160475585495357682065, −4.58313695244184402398234125030, −3.95854964706499578510428948131, −3.60796688827687417625254896040, −3.48964050612418510707632875488, −2.79593724146405308864962774541, −2.14528626285176441161686418426, −1.66203658693301302072053290249, −0.27122956379667115505741788950, 0.27122956379667115505741788950, 1.66203658693301302072053290249, 2.14528626285176441161686418426, 2.79593724146405308864962774541, 3.48964050612418510707632875488, 3.60796688827687417625254896040, 3.95854964706499578510428948131, 4.58313695244184402398234125030, 4.68532767160475585495357682065, 5.52315674764975307326848703611, 6.09256400202071666107587540515, 6.91907875715317599664719444575, 7.17333273231704129238110152830, 7.45016227287831454229440498385, 7.63935688462243273270055874365, 8.196739503927126331638371543454, 8.580426624740774002663258894686, 8.794012175979894117326943356371, 9.053432103286485331822425981631, 10.13213804910786260290176379548

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