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Properties

Label 4-28e4-1.1-c1e2-0-43
Degree $4$
Conductor $614656$
Sign $1$
Analytic cond. $39.1909$
Root an. cond. $2.50205$
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  − 2·3-s − 6·5-s + 3·9-s − 6·11-s + 12·15-s − 2·19-s − 6·23-s + 19·25-s − 10·27-s + 12·29-s + 8·31-s + 12·33-s + 2·37-s − 18·45-s − 6·53-s + 36·55-s + 4·57-s − 6·59-s + 6·61-s − 6·67-s + 12·69-s − 12·73-s − 38·75-s + 6·79-s + 20·81-s − 12·83-s − 24·87-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.68·5-s + 9-s − 1.80·11-s + 3.09·15-s − 0.458·19-s − 1.25·23-s + 19/5·25-s − 1.92·27-s + 2.22·29-s + 1.43·31-s + 2.08·33-s + 0.328·37-s − 2.68·45-s − 0.824·53-s + 4.85·55-s + 0.529·57-s − 0.781·59-s + 0.768·61-s − 0.733·67-s + 1.44·69-s − 1.40·73-s − 4.38·75-s + 0.675·79-s + 20/9·81-s − 1.31·83-s − 2.57·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(39.1909\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 614656,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.5.g_r
11$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_x
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.17.a_r
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2^2$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_bj
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_bh
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_abh
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.41.a_abi
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ax
61$C_2^2$ \( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_cv
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.g_db
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.71.a_afa
73$C_2^2$ \( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_er
79$C_2^2$ \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_dn
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2^2$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_fh
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−10.45064092404618624576041391837, −9.755912146266170157356787980229, −9.363460232001938192223691988724, −8.277330397860609314401042911877, −8.197497193922523645982764813321, −8.135106824139695214037094799330, −7.53938533565767705793031433471, −7.16918649032956188654591363765, −6.58641186022314235151630850492, −6.23313657320194833803067148870, −5.49229157437087386847296677144, −5.12899649451736412572202856689, −4.41482417676933497181119838178, −4.30963641674011181378805949560, −3.85077914416148281400819986736, −2.94662097412163382535566968564, −2.62881084871766801544384656958, −1.22966827996759562250840495380, 0, 0, 1.22966827996759562250840495380, 2.62881084871766801544384656958, 2.94662097412163382535566968564, 3.85077914416148281400819986736, 4.30963641674011181378805949560, 4.41482417676933497181119838178, 5.12899649451736412572202856689, 5.49229157437087386847296677144, 6.23313657320194833803067148870, 6.58641186022314235151630850492, 7.16918649032956188654591363765, 7.53938533565767705793031433471, 8.135106824139695214037094799330, 8.197497193922523645982764813321, 8.277330397860609314401042911877, 9.363460232001938192223691988724, 9.755912146266170157356787980229, 10.45064092404618624576041391837

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