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Properties

Label 4-39e2-1.1-c1e2-0-1
Degree $4$
Conductor $1521$
Sign $1$
Analytic cond. $0.0969802$
Root an. cond. $0.558047$
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  − 2·3-s + 4-s + 3·9-s − 2·12-s − 2·13-s − 3·16-s − 12·17-s + 10·25-s − 4·27-s + 12·29-s + 3·36-s + 4·39-s − 8·43-s + 6·48-s + 2·49-s + 24·51-s − 2·52-s + 12·53-s − 4·61-s − 7·64-s − 12·68-s − 20·75-s − 16·79-s + 5·81-s − 24·87-s + 10·100-s + 12·101-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 9-s − 0.577·12-s − 0.554·13-s − 3/4·16-s − 2.91·17-s + 2·25-s − 0.769·27-s + 2.22·29-s + 1/2·36-s + 0.640·39-s − 1.21·43-s + 0.866·48-s + 2/7·49-s + 3.36·51-s − 0.277·52-s + 1.64·53-s − 0.512·61-s − 7/8·64-s − 1.45·68-s − 2.30·75-s − 1.80·79-s + 5/9·81-s − 2.57·87-s + 100-s + 1.19·101-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.0969802\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1521,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4749534679\)
\(L(\frac12)\) \(\approx\) \(0.4749534679\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.41.a_abi
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.47.a_ade
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.71.a_afa
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \) 2.83.a_afy
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.89.a_afa
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−16.68934332216539481158586929182, −15.86429046282199755109551449657, −15.66671800979046012428842867774, −15.10589552252558216865316324167, −14.26255209441121924621344692766, −13.46495961562743261746007621118, −13.00091491120810677917759627526, −12.34089512118979564148306677207, −11.58516389668836280292342537270, −11.32665602148099343374542466665, −10.53459028746773639400487486073, −10.21653240500991125494313664343, −8.940367594292810735571675595475, −8.644174139697889043440842449786, −7.17583948104702118314526429719, −6.74370882529777620950526970910, −6.27213339115059087211963100445, −4.86905381270888678414367702714, −4.53114121658275230059828339895, −2.50907115140597324194211607825, 2.50907115140597324194211607825, 4.53114121658275230059828339895, 4.86905381270888678414367702714, 6.27213339115059087211963100445, 6.74370882529777620950526970910, 7.17583948104702118314526429719, 8.644174139697889043440842449786, 8.940367594292810735571675595475, 10.21653240500991125494313664343, 10.53459028746773639400487486073, 11.32665602148099343374542466665, 11.58516389668836280292342537270, 12.34089512118979564148306677207, 13.00091491120810677917759627526, 13.46495961562743261746007621118, 14.26255209441121924621344692766, 15.10589552252558216865316324167, 15.66671800979046012428842867774, 15.86429046282199755109551449657, 16.68934332216539481158586929182

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