Properties

Label 4-1386e2-1.1-c1e2-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·5-s − 5·7-s + 8-s − 3·10-s − 11-s + 12·13-s + 5·14-s − 16-s − 5·17-s − 6·19-s + 22-s + 5·23-s + 5·25-s − 12·26-s + 12·29-s − 4·31-s + 5·34-s − 15·35-s + 2·37-s + 6·38-s + 3·40-s − 10·41-s − 20·43-s − 5·46-s + 9·47-s + 18·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.34·5-s − 1.88·7-s + 0.353·8-s − 0.948·10-s − 0.301·11-s + 3.32·13-s + 1.33·14-s − 1/4·16-s − 1.21·17-s − 1.37·19-s + 0.213·22-s + 1.04·23-s + 25-s − 2.35·26-s + 2.22·29-s − 0.718·31-s + 0.857·34-s − 2.53·35-s + 0.328·37-s + 0.973·38-s + 0.474·40-s − 1.56·41-s − 3.04·43-s − 0.737·46-s + 1.31·47-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

\(L(1)\) \(\approx\) \(1.327016055\)
\(L(\frac12)\) \(\approx\) \(1.327016055\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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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

−9.916448842416488032827605078484, −9.118206553602416498153281073933, −9.006540446660709793182759469251, −8.747279706237932014664809760781, −8.400465084518878302233356933777, −8.105581476798569623220069197009, −7.04567856431553729244955602845, −6.70882626152288571719361777289, −6.59974913418605988984225055719, −6.20472428374815834006172146548, −5.83904825584983090881244272133, −5.45260091574327106129224170568, −4.54459492801154705183739351648, −4.34444960800290632302334056414, −3.36813368810486986421311984657, −3.35859804060016660428486740052, −2.69185802579767913790224877037, −1.86847297193809549682902764891, −1.42132518475133627083628390210, −0.55117606100553672993819145193, 0.55117606100553672993819145193, 1.42132518475133627083628390210, 1.86847297193809549682902764891, 2.69185802579767913790224877037, 3.35859804060016660428486740052, 3.36813368810486986421311984657, 4.34444960800290632302334056414, 4.54459492801154705183739351648, 5.45260091574327106129224170568, 5.83904825584983090881244272133, 6.20472428374815834006172146548, 6.59974913418605988984225055719, 6.70882626152288571719361777289, 7.04567856431553729244955602845, 8.105581476798569623220069197009, 8.400465084518878302233356933777, 8.747279706237932014664809760781, 9.006540446660709793182759469251, 9.118206553602416498153281073933, 9.916448842416488032827605078484

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