Properties

Label 4-968e2-1.1-c1e2-0-13
Degree $4$
Conductor $937024$
Sign $1$
Analytic cond. $59.7454$
Root an. cond. $2.78020$
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 − 2·7-s + 2·9-s − 8·13-s − 2·17-s − 2·19-s + 4·21-s + 6·23-s − 5·25-s − 6·27-s − 8·29-s + 14·31-s + 4·37-s + 16·39-s − 6·41-s − 12·43-s − 2·47-s − 6·49-s + 4·51-s − 4·53-s + 4·57-s − 20·59-s − 12·61-s − 4·63-s − 6·67-s − 12·69-s − 8·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 2/3·9-s − 2.21·13-s − 0.485·17-s − 0.458·19-s + 0.872·21-s + 1.25·23-s − 25-s − 1.15·27-s − 1.48·29-s + 2.51·31-s + 0.657·37-s + 2.56·39-s − 0.937·41-s − 1.82·43-s − 0.291·47-s − 6/7·49-s + 0.560·51-s − 0.549·53-s + 0.529·57-s − 2.60·59-s − 1.53·61-s − 0.503·63-s − 0.733·67-s − 1.44·69-s − 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(937024\)    =    \(2^{6} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(59.7454\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 937024,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
11 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_c
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.5.a_f
7$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_k
13$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bl
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.17.c_bj
19$C_4$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ag
23$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_by
29$D_{4}$ \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_cr
31$C_4$ \( 1 - 14 T + 106 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.31.ao_ec
37$D_{4}$ \( 1 - 4 T + 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_cv
41$D_{4}$ \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_ct
43$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_dy
47$D_{4}$ \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_by
53$D_{4}$ \( 1 + 4 T - 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_ap
59$D_{4}$ \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.59.u_hq
61$D_{4}$ \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_da
67$D_{4}$ \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_fi
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.71.i_gc
73$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_hi
79$D_{4}$ \( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_gw
83$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.83.c_bq
89$D_{4}$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.89.ac_ab
97$D_{4}$ \( 1 - 6 T + 123 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_et
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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

−9.931116979026305214120029100113, −9.430206392972404033826330277218, −9.223822505447146487165081620855, −8.650926008981888339656121298776, −7.938545450749815878903173333842, −7.53636199965051277700352969207, −7.38023217604878132352084690088, −6.65831896121089628634397318443, −6.28033577655893432972449925222, −6.16034493450490037523897119703, −5.41296221053125410294803803096, −4.92483661238660356728876322420, −4.64448235807922315633452748384, −4.26453706980168562601804730281, −3.17818398287563719065186625120, −3.06555448114678183175624842387, −2.16019315554053585259101657764, −1.52281091664324011096618361034, 0, 0, 1.52281091664324011096618361034, 2.16019315554053585259101657764, 3.06555448114678183175624842387, 3.17818398287563719065186625120, 4.26453706980168562601804730281, 4.64448235807922315633452748384, 4.92483661238660356728876322420, 5.41296221053125410294803803096, 6.16034493450490037523897119703, 6.28033577655893432972449925222, 6.65831896121089628634397318443, 7.38023217604878132352084690088, 7.53636199965051277700352969207, 7.938545450749815878903173333842, 8.650926008981888339656121298776, 9.223822505447146487165081620855, 9.430206392972404033826330277218, 9.931116979026305214120029100113

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