Properties

Label 4-931e2-1.1-c1e2-0-10
Degree $4$
Conductor $866761$
Sign $1$
Analytic cond. $55.2654$
Root an. cond. $2.72654$
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 + 3·3-s + 6·5-s − 3·6-s − 8-s + 4·9-s − 6·10-s − 5·11-s + 4·13-s + 18·15-s − 16-s + 7·17-s − 4·18-s − 2·19-s + 5·22-s − 6·23-s − 3·24-s + 17·25-s − 4·26-s + 6·27-s + 9·29-s − 18·30-s + 31-s + 6·32-s − 15·33-s − 7·34-s + 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 2.68·5-s − 1.22·6-s − 0.353·8-s + 4/3·9-s − 1.89·10-s − 1.50·11-s + 1.10·13-s + 4.64·15-s − 1/4·16-s + 1.69·17-s − 0.942·18-s − 0.458·19-s + 1.06·22-s − 1.25·23-s − 0.612·24-s + 17/5·25-s − 0.784·26-s + 1.15·27-s + 1.67·29-s − 3.28·30-s + 0.179·31-s + 1.06·32-s − 2.61·33-s − 1.20·34-s + 0.324·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 866761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 866761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(866761\)    =    \(7^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(55.2654\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 866761,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.040898048\)
\(L(\frac12)\) \(\approx\) \(4.040898048\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
3$C_4$ \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 17 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 9 T + 49 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 59 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 5 T + 85 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 2 T + 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 10 T + 115 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 15 T + 121 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 15 T + 193 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 14 T + 214 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 12 T + 217 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−9.993221323677472462720424300853, −9.909690724418764979675476989283, −9.296483892644072939166766323641, −9.251484283982329155265377174552, −8.465157922028266551654679290940, −8.385997594735006672065992933031, −8.054218178156319109016739957487, −7.71628845206791363792837877103, −6.58684002998486052032557530845, −6.53680405050374789868507128836, −6.15142754102135341081192219359, −5.47174431640610847000982248692, −5.15422408437747945404622014484, −4.68743584837260494509493933048, −3.53785132501114809093491309714, −3.25884414674399189401051979733, −2.70717446011079999376010246326, −2.07198691468930789539636512187, −1.96632893080951077507603908848, −0.993659226214071122921177239659, 0.993659226214071122921177239659, 1.96632893080951077507603908848, 2.07198691468930789539636512187, 2.70717446011079999376010246326, 3.25884414674399189401051979733, 3.53785132501114809093491309714, 4.68743584837260494509493933048, 5.15422408437747945404622014484, 5.47174431640610847000982248692, 6.15142754102135341081192219359, 6.53680405050374789868507128836, 6.58684002998486052032557530845, 7.71628845206791363792837877103, 8.054218178156319109016739957487, 8.385997594735006672065992933031, 8.465157922028266551654679290940, 9.251484283982329155265377174552, 9.296483892644072939166766323641, 9.909690724418764979675476989283, 9.993221323677472462720424300853

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