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Properties

Label 4-739328-1.1-c1e2-0-10
Degree $4$
Conductor $739328$
Sign $1$
Analytic cond. $47.1401$
Root an. cond. $2.62028$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 21·9-s + 4·11-s + 6·17-s − 2·19-s − 10·25-s + 54·27-s + 24·33-s − 24·41-s + 16·43-s − 13·49-s + 36·51-s − 12·57-s − 10·59-s + 14·67-s + 2·73-s − 60·75-s + 108·81-s + 12·83-s − 8·89-s − 12·97-s + 84·99-s + 10·107-s + 4·113-s − 10·121-s − 144·123-s + 127-s + ⋯
L(s)  = 1  + 3.46·3-s + 7·9-s + 1.20·11-s + 1.45·17-s − 0.458·19-s − 2·25-s + 10.3·27-s + 4.17·33-s − 3.74·41-s + 2.43·43-s − 1.85·49-s + 5.04·51-s − 1.58·57-s − 1.30·59-s + 1.71·67-s + 0.234·73-s − 6.92·75-s + 12·81-s + 1.31·83-s − 0.847·89-s − 1.21·97-s + 8.44·99-s + 0.966·107-s + 0.376·113-s − 0.909·121-s − 12.9·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(739328\)    =    \(2^{11} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(47.1401\)
Root analytic conductor: \(2.62028\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 739328,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.434052680\)
\(L(\frac12)\) \(\approx\) \(8.434052680\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )^{2} \) 2.3.ag_p
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.a_bx
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.41.y_is
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.59.k_fn
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.67.ao_hb
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.a_bq
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.73.ac_fr
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.89.i_hm
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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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

−8.197820813644998743406881037051, −7.943968172182858592403142779428, −7.74236222044461134007095453218, −7.28048593381483137900855724231, −6.60845811420238685670203067170, −6.39357456655427571414069968224, −5.46776906443032883183491997236, −4.82665740673161335065260299640, −4.08931083553499496867238827018, −3.66834687648261549407687934911, −3.63803508541238097399374767207, −2.94857221124987493994611399801, −2.43573806798025905726670590752, −1.66549496948467502043296521631, −1.52291178817025335850558392010, 1.52291178817025335850558392010, 1.66549496948467502043296521631, 2.43573806798025905726670590752, 2.94857221124987493994611399801, 3.63803508541238097399374767207, 3.66834687648261549407687934911, 4.08931083553499496867238827018, 4.82665740673161335065260299640, 5.46776906443032883183491997236, 6.39357456655427571414069968224, 6.60845811420238685670203067170, 7.28048593381483137900855724231, 7.74236222044461134007095453218, 7.943968172182858592403142779428, 8.197820813644998743406881037051

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