Properties

Label 4-1078e2-1.1-c1e2-0-26
Degree $4$
Conductor $1162084$
Sign $1$
Analytic cond. $74.0954$
Root an. cond. $2.93391$
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 + 2·3-s + 2·5-s − 2·6-s + 8-s + 3·9-s − 2·10-s − 11-s + 4·13-s + 4·15-s − 16-s − 3·18-s + 2·19-s + 22-s + 2·24-s + 5·25-s − 4·26-s + 10·27-s + 12·29-s − 4·30-s − 4·31-s − 2·33-s − 2·37-s − 2·38-s + 8·39-s + 2·40-s + 16·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 0.894·5-s − 0.816·6-s + 0.353·8-s + 9-s − 0.632·10-s − 0.301·11-s + 1.10·13-s + 1.03·15-s − 1/4·16-s − 0.707·18-s + 0.458·19-s + 0.213·22-s + 0.408·24-s + 25-s − 0.784·26-s + 1.92·27-s + 2.22·29-s − 0.730·30-s − 0.718·31-s − 0.348·33-s − 0.328·37-s − 0.324·38-s + 1.28·39-s + 0.316·40-s + 2.49·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1162084\)    =    \(2^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(74.0954\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1078} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1162084,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.333937939\)
\(L(\frac12)\) \(\approx\) \(3.333937939\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
11$C_2$ \( 1 + T + T^{2} \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−9.878480197869522918873785902868, −9.569132257440736583791078470820, −9.277313174607398447291313851018, −8.833126940553592347636614710483, −8.528670673808767553411287838081, −8.185673751954384129227300022030, −7.78695518664863755018312269860, −7.29904018668731221920527153075, −6.70386142176735825779759230728, −6.58148929550446217701169107224, −5.68212035210045430791930913610, −5.63999097904389062575008372897, −4.80590847402103469607571089897, −4.21956228880155036300121981627, −4.02676480271910917857165975411, −2.98330363049331644095271697287, −2.79555753753060316965791167108, −2.30660920713042264604131737042, −1.21874606892508270594140318378, −1.10051110655016347381152921607, 1.10051110655016347381152921607, 1.21874606892508270594140318378, 2.30660920713042264604131737042, 2.79555753753060316965791167108, 2.98330363049331644095271697287, 4.02676480271910917857165975411, 4.21956228880155036300121981627, 4.80590847402103469607571089897, 5.63999097904389062575008372897, 5.68212035210045430791930913610, 6.58148929550446217701169107224, 6.70386142176735825779759230728, 7.29904018668731221920527153075, 7.78695518664863755018312269860, 8.185673751954384129227300022030, 8.528670673808767553411287838081, 8.833126940553592347636614710483, 9.277313174607398447291313851018, 9.569132257440736583791078470820, 9.878480197869522918873785902868

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