Properties

Label 4-1386e2-1.1-c1e2-0-13
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 + 2·5-s + 4·7-s + 8-s − 2·10-s − 11-s + 4·13-s − 4·14-s − 16-s − 17-s + 3·19-s + 22-s + 23-s + 5·25-s − 4·26-s − 2·29-s + 2·31-s + 34-s + 8·35-s + 5·37-s − 3·38-s + 2·40-s − 20·41-s + 2·43-s − 46-s − 7·47-s + 9·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.894·5-s + 1.51·7-s + 0.353·8-s − 0.632·10-s − 0.301·11-s + 1.10·13-s − 1.06·14-s − 1/4·16-s − 0.242·17-s + 0.688·19-s + 0.213·22-s + 0.208·23-s + 25-s − 0.784·26-s − 0.371·29-s + 0.359·31-s + 0.171·34-s + 1.35·35-s + 0.821·37-s − 0.486·38-s + 0.316·40-s − 3.12·41-s + 0.304·43-s − 0.147·46-s − 1.02·47-s + 9/7·49-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

\(L(1)\) \(\approx\) \(2.369454517\)
\(L(\frac12)\) \(\approx\) \(2.369454517\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 8 T - 9 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−9.889477312920301324964904860507, −9.311935652411048986490247624879, −8.863239420121568335880281049672, −8.597306716603120223905779381596, −8.256929195971284022731157773286, −7.970832409439989107870165291578, −7.38597638897484079951488206015, −7.09309338290594131659067586554, −6.38491491524175610387571592154, −6.24827220697949292882323586827, −5.53491908217530440620002485895, −5.17793386155280639859768629554, −4.76409147569165576349404978167, −4.51173447792363262900651372980, −3.50768630341792236030428700600, −3.34972173498370529359168213142, −2.40889638730991916345205184992, −1.83489091306361487411222903180, −1.47521264060918931232613496761, −0.76425809182926482829844308124, 0.76425809182926482829844308124, 1.47521264060918931232613496761, 1.83489091306361487411222903180, 2.40889638730991916345205184992, 3.34972173498370529359168213142, 3.50768630341792236030428700600, 4.51173447792363262900651372980, 4.76409147569165576349404978167, 5.17793386155280639859768629554, 5.53491908217530440620002485895, 6.24827220697949292882323586827, 6.38491491524175610387571592154, 7.09309338290594131659067586554, 7.38597638897484079951488206015, 7.970832409439989107870165291578, 8.256929195971284022731157773286, 8.597306716603120223905779381596, 8.863239420121568335880281049672, 9.311935652411048986490247624879, 9.889477312920301324964904860507

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