Properties

Label 4-4864e2-1.1-c1e2-0-9
Degree $4$
Conductor $23658496$
Sign $1$
Analytic cond. $1508.48$
Root an. cond. $6.23211$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 6·9-s − 6·11-s − 6·17-s + 2·19-s − 7·25-s − 4·27-s − 24·33-s − 12·41-s − 2·43-s − 11·49-s − 24·51-s + 8·57-s − 12·59-s + 8·67-s + 2·73-s − 28·75-s − 37·81-s − 24·83-s + 8·97-s − 36·99-s − 12·107-s − 36·113-s + 5·121-s − 48·123-s + 127-s − 8·129-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s − 1.80·11-s − 1.45·17-s + 0.458·19-s − 7/5·25-s − 0.769·27-s − 4.17·33-s − 1.87·41-s − 0.304·43-s − 1.57·49-s − 3.36·51-s + 1.05·57-s − 1.56·59-s + 0.977·67-s + 0.234·73-s − 3.23·75-s − 4.11·81-s − 2.63·83-s + 0.812·97-s − 3.61·99-s − 1.16·107-s − 3.38·113-s + 5/11·121-s − 4.32·123-s + 0.0887·127-s − 0.704·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23658496\)    =    \(2^{16} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1508.48\)
Root analytic conductor: \(6.23211\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4864} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 23658496,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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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

−8.074165407662890807330090562647, −7.967601908046062758498727639263, −7.54154771182837907562951747214, −7.28084420153821384462264955127, −6.68402182678737925633599303039, −6.49196180865455768295195716946, −5.75619920146915586569592688283, −5.59503632385320594669237283970, −5.00873351546071334996561317597, −4.79398157420474018498834957177, −4.07981947445611839833999123970, −3.92143101781958611706780236892, −3.20456748057452351457019907730, −3.17174252380299729323959684818, −2.56569723035598843527809779715, −2.43476512807659373600785578284, −1.78284891188202082820909407697, −1.59507462490591482548231560399, 0, 0, 1.59507462490591482548231560399, 1.78284891188202082820909407697, 2.43476512807659373600785578284, 2.56569723035598843527809779715, 3.17174252380299729323959684818, 3.20456748057452351457019907730, 3.92143101781958611706780236892, 4.07981947445611839833999123970, 4.79398157420474018498834957177, 5.00873351546071334996561317597, 5.59503632385320594669237283970, 5.75619920146915586569592688283, 6.49196180865455768295195716946, 6.68402182678737925633599303039, 7.28084420153821384462264955127, 7.54154771182837907562951747214, 7.967601908046062758498727639263, 8.074165407662890807330090562647

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