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Properties

Label 4-987696-1.1-c1e2-0-4
Degree $4$
Conductor $987696$
Sign $1$
Analytic cond. $62.9763$
Root an. cond. $2.81704$
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  − 2·2-s + 2·3-s + 2·4-s + 2·5-s − 4·6-s + 3·9-s − 4·10-s + 4·12-s + 4·15-s − 4·16-s + 6·17-s − 6·18-s − 19-s + 4·20-s − 7·25-s + 4·27-s − 8·30-s + 4·31-s + 8·32-s − 12·34-s + 6·36-s + 2·38-s + 6·45-s − 8·48-s − 5·49-s + 14·50-s + 12·51-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 4-s + 0.894·5-s − 1.63·6-s + 9-s − 1.26·10-s + 1.15·12-s + 1.03·15-s − 16-s + 1.45·17-s − 1.41·18-s − 0.229·19-s + 0.894·20-s − 7/5·25-s + 0.769·27-s − 1.46·30-s + 0.718·31-s + 1.41·32-s − 2.05·34-s + 36-s + 0.324·38-s + 0.894·45-s − 1.15·48-s − 5/7·49-s + 1.97·50-s + 1.68·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(987696\)    =    \(2^{4} \cdot 3^{2} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(62.9763\)
Root analytic conductor: \(2.81704\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 987696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.972938037\)
\(L(\frac12)\) \(\approx\) \(1.972938037\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.a_dh
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.a_dh
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−8.097404073348217758325727576066, −8.062798749262723709584757589645, −7.42319981131404835915630105083, −7.05312905485418028387733633198, −6.54349707111784488533663869338, −6.06737359159706268125478251824, −5.51581757373263673213280839728, −5.02993077790447449830764321556, −4.36012986824685430367932721730, −3.75304137241909978574120299805, −3.31734048068229197388256659419, −2.48928634219704741195361825764, −2.13307257724406321246824925475, −1.55575231856794394588809977183, −0.801024832446311832814916696585, 0.801024832446311832814916696585, 1.55575231856794394588809977183, 2.13307257724406321246824925475, 2.48928634219704741195361825764, 3.31734048068229197388256659419, 3.75304137241909978574120299805, 4.36012986824685430367932721730, 5.02993077790447449830764321556, 5.51581757373263673213280839728, 6.06737359159706268125478251824, 6.54349707111784488533663869338, 7.05312905485418028387733633198, 7.42319981131404835915630105083, 8.062798749262723709584757589645, 8.097404073348217758325727576066

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