Properties

Label 4-1386e2-1.1-c1e2-0-14
Degree $4$
Conductor $1920996$
Sign $1$
Analytic cond. $122.484$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 5-s − 5·7-s − 8-s − 10-s − 11-s + 4·13-s − 5·14-s − 16-s + 5·17-s + 6·19-s − 22-s + 7·23-s + 5·25-s + 4·26-s + 16·29-s − 10·31-s + 5·34-s + 5·35-s + 8·37-s + 6·38-s + 40-s − 14·41-s + 8·43-s + 7·46-s − 47-s + 18·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.10·13-s − 1.33·14-s − 1/4·16-s + 1.21·17-s + 1.37·19-s − 0.213·22-s + 1.45·23-s + 25-s + 0.784·26-s + 2.97·29-s − 1.79·31-s + 0.857·34-s + 0.845·35-s + 1.31·37-s + 0.973·38-s + 0.158·40-s − 2.18·41-s + 1.21·43-s + 1.03·46-s − 0.145·47-s + 18/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1920996\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(122.484\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1386} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1920996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.773316029\)
\(L(\frac12)\) \(\approx\) \(2.773316029\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
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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

−9.722942406712210365253753669600, −9.384282573028651789534262960905, −8.912745131488240073196171624940, −8.803437196083605841855088858414, −8.026861769763379612240257243805, −7.83387624338365706137608638082, −7.16336257771782792552786202175, −6.78158456778465121390164540720, −6.55421423571550900349679512308, −6.10146751378147095158105901835, −5.50847474699024380613983410841, −5.21766004599126879123267303568, −4.84087452066317297379660552526, −4.05529041732218200274643882425, −3.64370442573907985162946833372, −3.21729307601242294853268920414, −3.03836291788202732639667413891, −2.46910547395423037472204259919, −1.07291782668336743015093228662, −0.77161620851336411024148920747, 0.77161620851336411024148920747, 1.07291782668336743015093228662, 2.46910547395423037472204259919, 3.03836291788202732639667413891, 3.21729307601242294853268920414, 3.64370442573907985162946833372, 4.05529041732218200274643882425, 4.84087452066317297379660552526, 5.21766004599126879123267303568, 5.50847474699024380613983410841, 6.10146751378147095158105901835, 6.55421423571550900349679512308, 6.78158456778465121390164540720, 7.16336257771782792552786202175, 7.83387624338365706137608638082, 8.026861769763379612240257243805, 8.803437196083605841855088858414, 8.912745131488240073196171624940, 9.384282573028651789534262960905, 9.722942406712210365253753669600

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