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Properties

Label 4-238e2-1.1-c1e2-0-7
Degree $4$
Conductor $56644$
Sign $1$
Analytic cond. $3.61167$
Root an. cond. $1.37856$
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  + 2-s + 3-s + 2·5-s + 6-s − 4·7-s − 8-s + 3·9-s + 2·10-s + 11-s + 10·13-s − 4·14-s + 2·15-s − 16-s + 17-s + 3·18-s − 6·19-s − 4·21-s + 22-s − 24-s + 5·25-s + 10·26-s + 8·27-s − 12·29-s + 2·30-s − 4·31-s + 33-s + 34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.894·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 9-s + 0.632·10-s + 0.301·11-s + 2.77·13-s − 1.06·14-s + 0.516·15-s − 1/4·16-s + 0.242·17-s + 0.707·18-s − 1.37·19-s − 0.872·21-s + 0.213·22-s − 0.204·24-s + 25-s + 1.96·26-s + 1.53·27-s − 2.22·29-s + 0.365·30-s − 0.718·31-s + 0.174·33-s + 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56644\)    =    \(2^{2} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(3.61167\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 56644,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.593236571\)
\(L(\frac12)\) \(\approx\) \(2.593236571\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
17$C_2$ \( 1 - T + T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_ak
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.13.ak_bz
19$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_r
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bb
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_abr
53$C_2^2$ \( 1 - 7 T - 4 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.53.ah_ae
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_dh
61$C_2^2$ \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_df
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_acl
71$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.71.ao_hj
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2^2$ \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.79.d_acs
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2^2$ \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_adk
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.97.y_na
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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

−12.79348523722218745012008879074, −12.14889593285371374490426980238, −11.38568417233479250575594470800, −10.86632616735349538675347233578, −10.45638914810502961483803892897, −10.00201167277202756392568399579, −9.335594379323794776518501887713, −9.155065888105107479082298857683, −8.389557331501682570663053311767, −8.354900601076673600442950202119, −7.01319826621547536290420362109, −6.68000034832818214163693126838, −6.44455182547807385835845326024, −5.65351290544615791559171850687, −5.35512925343280538803065050888, −4.15259246698193771552591432243, −3.61751441304990806598001717871, −3.52567205019278328548120295689, −2.34862110339716896619279711302, −1.41122341362382120862116286465, 1.41122341362382120862116286465, 2.34862110339716896619279711302, 3.52567205019278328548120295689, 3.61751441304990806598001717871, 4.15259246698193771552591432243, 5.35512925343280538803065050888, 5.65351290544615791559171850687, 6.44455182547807385835845326024, 6.68000034832818214163693126838, 7.01319826621547536290420362109, 8.354900601076673600442950202119, 8.389557331501682570663053311767, 9.155065888105107479082298857683, 9.335594379323794776518501887713, 10.00201167277202756392568399579, 10.45638914810502961483803892897, 10.86632616735349538675347233578, 11.38568417233479250575594470800, 12.14889593285371374490426980238, 12.79348523722218745012008879074

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