Properties

Label 4-42e2-1.1-c1e2-0-1
Degree $4$
Conductor $1764$
Sign $1$
Analytic cond. $0.112474$
Root an. cond. $0.579112$
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-s + 3-s − 5-s − 6-s + 7-s + 8-s + 10-s − 5·11-s − 14-s − 15-s − 16-s + 4·17-s − 8·19-s + 21-s + 5·22-s + 4·23-s + 24-s + 5·25-s − 27-s − 10·29-s + 30-s − 3·31-s − 5·33-s − 4·34-s − 35-s + 4·37-s + 8·38-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 + 0.316·10-s − 1.50·11-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.970·17-s − 1.83·19-s + 0.218·21-s + 1.06·22-s + 0.834·23-s + 0.204·24-s + 25-s − 0.192·27-s − 1.85·29-s + 0.182·30-s − 0.538·31-s − 0.870·33-s − 0.685·34-s − 0.169·35-s + 0.657·37-s + 1.29·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.112474\)
Root analytic conductor: \(0.579112\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{42} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1764,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4773693664\)
\(L(\frac12)\) \(\approx\) \(0.4773693664\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−16.48137555834193247014456911537, −15.83619965200438823876216004821, −15.04713662038036657454485295735, −14.80964254972782301167757611807, −14.28937209773509806193891437398, −13.14936952808423716407580636833, −13.07144868289491506618070102816, −12.41612234657133509592482402351, −11.30334526510677373324538947150, −10.93716567284204152926305098868, −10.30103747365264547233487855111, −9.631223654733318138518083191236, −8.576514985716478914628933745839, −8.566127481624805597415792116047, −7.56606074256781909532176097676, −7.23470899635064634139809584743, −5.81813496944578039373917664188, −4.95246024856700034661833979297, −3.82388470647963604033512192263, −2.46289452936338301129272717961, 2.46289452936338301129272717961, 3.82388470647963604033512192263, 4.95246024856700034661833979297, 5.81813496944578039373917664188, 7.23470899635064634139809584743, 7.56606074256781909532176097676, 8.566127481624805597415792116047, 8.576514985716478914628933745839, 9.631223654733318138518083191236, 10.30103747365264547233487855111, 10.93716567284204152926305098868, 11.30334526510677373324538947150, 12.41612234657133509592482402351, 13.07144868289491506618070102816, 13.14936952808423716407580636833, 14.28937209773509806193891437398, 14.80964254972782301167757611807, 15.04713662038036657454485295735, 15.83619965200438823876216004821, 16.48137555834193247014456911537

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