Properties

Label 4-770e2-1.1-c1e2-0-5
Degree $4$
Conductor $592900$
Sign $1$
Analytic cond. $37.8038$
Root an. cond. $2.47961$
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 − 3-s + 5-s − 6-s − 7-s − 8-s + 3·9-s + 10-s − 11-s + 10·13-s − 14-s − 15-s − 16-s + 3·18-s − 2·19-s + 21-s − 22-s − 6·23-s + 24-s + 10·26-s − 8·27-s − 6·29-s − 30-s − 8·31-s + 33-s − 35-s + 4·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 9-s + 0.316·10-s − 0.301·11-s + 2.77·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.707·18-s − 0.458·19-s + 0.218·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s + 1.96·26-s − 1.53·27-s − 1.11·29-s − 0.182·30-s − 1.43·31-s + 0.174·33-s − 0.169·35-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(592900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(37.8038\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 592900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.375342940\)
\(L(\frac12)\) \(\approx\) \(2.375342940\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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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

−10.71756050917693295279591298016, −9.951037129101681170483018986788, −9.870660247788252462588946617189, −9.344295859068220283007987475760, −8.874547167063167090760489988417, −8.218692980235785726831868485222, −8.203327279379770962016061349061, −7.25092236668523456987634725760, −7.05136444972120447104244892023, −6.19920218164893090584784692115, −6.18311375864794947797885980045, −5.60161601625575174647572375068, −5.47067773703904518483074684020, −4.54365015614014438407332393867, −4.07117250307312625726916207066, −3.62014481981674702749273047087, −3.44801337292150060346098469544, −2.13655579088538637475744861357, −1.79840246070835368576109739255, −0.74616381343278522036145526268, 0.74616381343278522036145526268, 1.79840246070835368576109739255, 2.13655579088538637475744861357, 3.44801337292150060346098469544, 3.62014481981674702749273047087, 4.07117250307312625726916207066, 4.54365015614014438407332393867, 5.47067773703904518483074684020, 5.60161601625575174647572375068, 6.18311375864794947797885980045, 6.19920218164893090584784692115, 7.05136444972120447104244892023, 7.25092236668523456987634725760, 8.203327279379770962016061349061, 8.218692980235785726831868485222, 8.874547167063167090760489988417, 9.344295859068220283007987475760, 9.870660247788252462588946617189, 9.951037129101681170483018986788, 10.71756050917693295279591298016

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