Properties

Label 4-930e2-1.1-c1e2-0-0
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
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  − 4-s − 2·5-s − 9-s − 6·11-s + 16-s + 2·19-s + 2·20-s − 25-s − 4·29-s − 2·31-s + 36-s − 20·41-s + 6·44-s + 2·45-s + 13·49-s + 12·55-s + 12·59-s − 4·61-s − 64-s − 10·71-s − 2·76-s − 6·79-s − 2·80-s + 81-s − 2·89-s − 4·95-s + 6·99-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s − 1/3·9-s − 1.80·11-s + 1/4·16-s + 0.458·19-s + 0.447·20-s − 1/5·25-s − 0.742·29-s − 0.359·31-s + 1/6·36-s − 3.12·41-s + 0.904·44-s + 0.298·45-s + 13/7·49-s + 1.61·55-s + 1.56·59-s − 0.512·61-s − 1/8·64-s − 1.18·71-s − 0.229·76-s − 0.675·79-s − 0.223·80-s + 1/9·81-s − 0.211·89-s − 0.410·95-s + 0.603·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{930} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3377973826\)
\(L(\frac12)\) \(\approx\) \(0.3377973826\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 81 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−10.44008843758896986894137273194, −9.860805618875227593481910657206, −9.545010160273345635842361833700, −8.800218478491754257952755942119, −8.627607386761118794623801111332, −8.154227292792793092578435497507, −7.86416968838300278318250902741, −7.30237574232669532135802547708, −7.14030689535698297919481576656, −6.48148449414858039901327596220, −5.69711276393135988105786699428, −5.43887489988290248425580158710, −5.12899848982183335966401969341, −4.52910691332778202043452752015, −3.92168080914936600301426973713, −3.54288568469892368614829201748, −2.92225776139589165403907765123, −2.40531935874045117295226274603, −1.51866075989609014854314795593, −0.27446055720847916293724345868, 0.27446055720847916293724345868, 1.51866075989609014854314795593, 2.40531935874045117295226274603, 2.92225776139589165403907765123, 3.54288568469892368614829201748, 3.92168080914936600301426973713, 4.52910691332778202043452752015, 5.12899848982183335966401969341, 5.43887489988290248425580158710, 5.69711276393135988105786699428, 6.48148449414858039901327596220, 7.14030689535698297919481576656, 7.30237574232669532135802547708, 7.86416968838300278318250902741, 8.154227292792793092578435497507, 8.627607386761118794623801111332, 8.800218478491754257952755942119, 9.545010160273345635842361833700, 9.860805618875227593481910657206, 10.44008843758896986894137273194

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