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Properties

Label 4-273e2-1.1-c1e2-0-8
Degree $4$
Conductor $74529$
Sign $1$
Analytic cond. $4.75203$
Root an. cond. $1.47645$
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  + 2·2-s − 2·3-s + 4-s − 4·6-s + 2·7-s + 3·9-s + 4·11-s − 2·12-s − 2·13-s + 4·14-s + 16-s + 4·17-s + 6·18-s − 4·21-s + 8·22-s + 8·23-s − 10·25-s − 4·26-s − 4·27-s + 2·28-s + 4·29-s − 8·31-s − 2·32-s − 8·33-s + 8·34-s + 3·36-s − 4·37-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.63·6-s + 0.755·7-s + 9-s + 1.20·11-s − 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s + 1.41·18-s − 0.872·21-s + 1.70·22-s + 1.66·23-s − 2·25-s − 0.784·26-s − 0.769·27-s + 0.377·28-s + 0.742·29-s − 1.43·31-s − 0.353·32-s − 1.39·33-s + 1.37·34-s + 1/2·36-s − 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74529\)    =    \(3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(4.75203\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 74529,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.241654447\)
\(L(\frac12)\) \(\approx\) \(2.241654447\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_d
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_be
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_cc
29$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_be
31$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bu
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_bu
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.41.a_by
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_cs
47$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_es
53$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_da
59$D_{4}$ \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_by
61$D_{4}$ \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_ew
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.71.abc_na
73$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_eo
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$D_{4}$ \( 1 - 4 T - 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_abe
89$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_co
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−12.11879618305232732790366468940, −11.91778421432812333688761023369, −11.29518608843858614815163999582, −10.93021901693832264467984756681, −10.55765662806467945537028057206, −9.868129868172996668906535224299, −9.226081946387546687606087650199, −9.083653609151694394209207435823, −8.035619877352594052656783644743, −7.54413780415935792002410909489, −7.13600036933454380613210868054, −6.50523871037060419535490232534, −5.71245602426191992357572839372, −5.62715207324924267832557476789, −4.85387741242917259082489195629, −4.63869215328836944881622050919, −3.81564192422094497201229732335, −3.54290927457861224230656796089, −2.14157049850050474625482236726, −1.13885815283392249125382284398, 1.13885815283392249125382284398, 2.14157049850050474625482236726, 3.54290927457861224230656796089, 3.81564192422094497201229732335, 4.63869215328836944881622050919, 4.85387741242917259082489195629, 5.62715207324924267832557476789, 5.71245602426191992357572839372, 6.50523871037060419535490232534, 7.13600036933454380613210868054, 7.54413780415935792002410909489, 8.035619877352594052656783644743, 9.083653609151694394209207435823, 9.226081946387546687606087650199, 9.868129868172996668906535224299, 10.55765662806467945537028057206, 10.93021901693832264467984756681, 11.29518608843858614815163999582, 11.91778421432812333688761023369, 12.11879618305232732790366468940

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