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Properties

Label 4-82e2-1.1-c1e2-0-1
Degree $4$
Conductor $6724$
Sign $1$
Analytic cond. $0.428728$
Root an. cond. $0.809180$
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  − 2·2-s + 3·4-s − 4·8-s + 4·9-s + 5·16-s − 8·18-s − 10·25-s + 16·31-s − 6·32-s + 12·36-s + 16·37-s − 6·41-s − 8·43-s − 4·49-s + 20·50-s − 24·59-s + 4·61-s − 32·62-s + 7·64-s − 16·72-s − 8·73-s − 32·74-s + 7·81-s + 12·82-s + 24·83-s + 16·86-s + 8·98-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 4/3·9-s + 5/4·16-s − 1.88·18-s − 2·25-s + 2.87·31-s − 1.06·32-s + 2·36-s + 2.63·37-s − 0.937·41-s − 1.21·43-s − 4/7·49-s + 2.82·50-s − 3.12·59-s + 0.512·61-s − 4.06·62-s + 7/8·64-s − 1.88·72-s − 0.936·73-s − 3.71·74-s + 7/9·81-s + 1.32·82-s + 2.63·83-s + 1.72·86-s + 0.808·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6724\)    =    \(2^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(0.428728\)
Root analytic conductor: \(0.809180\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6724,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5332554831\)
\(L(\frac12)\) \(\approx\) \(0.5332554831\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
41$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.11.a_au
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.47.a_abs
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.67.a_bc
71$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.71.a_ado
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \) 2.79.a_afk
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.a_dq
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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

−15.04125845260439248814778886338, −13.99099649675383920407175138354, −13.46203202579228341404552085428, −13.05546580731004234433260748138, −12.04233198463357823386335406128, −11.98075634686149634766223001848, −11.24693991564168238771013131117, −10.57895286184314210273779158945, −9.955900386555452119830042268101, −9.745882746813284949042619037994, −9.231876366086698723292105083584, −8.160726857578117042551346737256, −7.979709927964138581800897823776, −7.37453055558091235534647520739, −6.36239357201780679996803335740, −6.28302350935325781901961139120, −4.86979712975079139223570549762, −4.03604900400171253358567908044, −2.74281168846530704326542133326, −1.47997097453131561975038860063, 1.47997097453131561975038860063, 2.74281168846530704326542133326, 4.03604900400171253358567908044, 4.86979712975079139223570549762, 6.28302350935325781901961139120, 6.36239357201780679996803335740, 7.37453055558091235534647520739, 7.979709927964138581800897823776, 8.160726857578117042551346737256, 9.231876366086698723292105083584, 9.745882746813284949042619037994, 9.955900386555452119830042268101, 10.57895286184314210273779158945, 11.24693991564168238771013131117, 11.98075634686149634766223001848, 12.04233198463357823386335406128, 13.05546580731004234433260748138, 13.46203202579228341404552085428, 13.99099649675383920407175138354, 15.04125845260439248814778886338

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