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Properties

Label 4-39e2-1.1-c20e2-0-0
Degree $4$
Conductor $1521$
Sign $1$
Analytic cond. $9775.34$
Root an. cond. $9.94335$
Motivic weight $20$
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  − 1.18e5·3-s + 1.87e6·4-s + 1.04e10·9-s − 2.21e11·12-s − 2.75e11·13-s + 2.40e12·16-s + 1.75e14·25-s − 8.23e14·27-s + 1.95e16·36-s + 3.25e16·39-s + 8.61e16·43-s − 2.83e17·48-s + 1.59e17·49-s − 5.16e17·52-s + 2.81e18·61-s + 2.43e18·64-s − 2.07e19·75-s − 3.77e19·79-s + 6.07e19·81-s + 3.28e20·100-s + 4.14e20·103-s − 1.54e21·108-s − 2.88e21·117-s − 1.24e21·121-s + 127-s − 1.01e22·129-s + 131-s + ⋯
L(s)  = 1  − 2·3-s + 1.78·4-s + 3·9-s − 3.56·12-s − 2·13-s + 2.18·16-s + 1.84·25-s − 4·27-s + 5.35·36-s + 4·39-s + 3.98·43-s − 4.37·48-s + 2·49-s − 3.56·52-s + 3.94·61-s + 2.11·64-s − 3.68·75-s − 3.98·79-s + 5·81-s + 3.28·100-s + 3.08·103-s − 7.13·108-s − 6·117-s − 1.84·121-s − 7.97·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+10)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(9775.34\)
Root analytic conductor: \(9.94335\)
Motivic weight: \(20\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1521,\ (\ :10, 10),\ 1)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(3.484655434\)
\(L(\frac12)\) \(\approx\) \(3.484655434\)
\(L(11)\) 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)$
bad3$C_1$ \( ( 1 + p^{10} T )^{2} \)
13$C_1$ \( ( 1 + p^{10} T )^{2} \)
good2$C_2^2$ \( 1 - 1871527 T^{2} + p^{40} T^{4} \)
5$C_2^2$ \( 1 - 175670513317726 T^{2} + p^{40} T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
11$C_2^2$ \( 1 + \)\(12\!\cdots\!98\)\( T^{2} + p^{40} T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
41$C_2^2$ \( 1 + \)\(14\!\cdots\!98\)\( T^{2} + p^{40} T^{4} \)
43$C_2$ \( ( 1 - 43068829770054002 T + p^{20} T^{2} )^{2} \)
47$C_2^2$ \( 1 + \)\(52\!\cdots\!98\)\( T^{2} + p^{40} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
59$C_2^2$ \( 1 + \)\(17\!\cdots\!98\)\( T^{2} + p^{40} T^{4} \)
61$C_2$ \( ( 1 - 1405594259224342702 T + p^{20} T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
71$C_2^2$ \( 1 + \)\(16\!\cdots\!98\)\( T^{2} + p^{40} T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
79$C_2$ \( ( 1 + 18878509775468868098 T + p^{20} T^{2} )^{2} \)
83$C_2^2$ \( 1 + \)\(32\!\cdots\!98\)\( T^{2} + p^{40} T^{4} \)
89$C_2^2$ \( 1 - \)\(19\!\cdots\!02\)\( T^{2} + p^{40} T^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - p^{10} T )^{2}( 1 + p^{10} T )^{2} \)
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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

−12.24901008431176882182156794058, −11.56242198629811578471781258624, −11.41271275369253319080207417581, −10.64872987631109459505759403085, −10.33529605029272483892534303945, −9.876425878429008711396562548027, −8.942627248612481802023352367073, −7.62328804754574669897991280947, −7.19380858517833793045075712199, −7.05503734298843950037214512094, −6.35651260215618365217087106407, −5.60958087721581586950779415485, −5.41253543338752230801863637254, −4.54428722291209977965416011785, −3.95793763730484230873616701126, −2.62536833433110858924810441006, −2.46654956467635689318487867885, −1.61240563253564136533352956791, −0.72674229578713030575416771851, −0.68199805912332616072460089454, 0.68199805912332616072460089454, 0.72674229578713030575416771851, 1.61240563253564136533352956791, 2.46654956467635689318487867885, 2.62536833433110858924810441006, 3.95793763730484230873616701126, 4.54428722291209977965416011785, 5.41253543338752230801863637254, 5.60958087721581586950779415485, 6.35651260215618365217087106407, 7.05503734298843950037214512094, 7.19380858517833793045075712199, 7.62328804754574669897991280947, 8.942627248612481802023352367073, 9.876425878429008711396562548027, 10.33529605029272483892534303945, 10.64872987631109459505759403085, 11.41271275369253319080207417581, 11.56242198629811578471781258624, 12.24901008431176882182156794058

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