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Properties

Label 4-465e2-1.1-c1e2-0-5
Degree $4$
Conductor $216225$
Sign $1$
Analytic cond. $13.7866$
Root an. cond. $1.92692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 4-s − 2·5-s − 4·6-s − 4·7-s + 3·9-s + 4·10-s + 2·12-s − 8·13-s + 8·14-s − 4·15-s + 16-s − 8·17-s − 6·18-s − 2·20-s − 8·21-s − 12·23-s + 3·25-s + 16·26-s + 4·27-s − 4·28-s − 4·29-s + 8·30-s + 2·31-s + 2·32-s + 16·34-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s − 1.51·7-s + 9-s + 1.26·10-s + 0.577·12-s − 2.21·13-s + 2.13·14-s − 1.03·15-s + 1/4·16-s − 1.94·17-s − 1.41·18-s − 0.447·20-s − 1.74·21-s − 2.50·23-s + 3/5·25-s + 3.13·26-s + 0.769·27-s − 0.755·28-s − 0.742·29-s + 1.46·30-s + 0.359·31-s + 0.353·32-s + 2.74·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(216225\)    =    \(3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(13.7866\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 216225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_d
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_q
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bo
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$D_{4}$ \( 1 + 4 T + 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_bs
37$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \) 2.37.a_cu
41$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_da
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.43.a_abq
47$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_dm
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.53.i_es
59$D_{4}$ \( 1 - 8 T + 116 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_em
61$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_cc
67$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_cu
71$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.71.a_bs
73$D_{4}$ \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_cm
79$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_es
83$D_{4}$ \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_go
89$D_{4}$ \( 1 + 4 T + 132 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_fc
97$D_{4}$ \( 1 - 20 T + 262 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.97.au_kc
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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

−10.32235555061267466180822521169, −10.18009326252125165046068149465, −9.741129329602803732537056492888, −9.444181284140794991678704857799, −9.077820780077449798814052905039, −8.706328387019095597966801261923, −8.043358386652882921381706620166, −7.909791602786781256767800212665, −7.36707277876973272180811318595, −6.90386867118158446352776102535, −6.42990980240923616179524075766, −5.87664044298959329900745955648, −4.75017997526825188335544832936, −4.40631948621539143839861037043, −3.83039348159519505383301947087, −3.10080138876143517095807165572, −2.55413117709499278887580036066, −1.95037852596544015350199669678, 0, 0, 1.95037852596544015350199669678, 2.55413117709499278887580036066, 3.10080138876143517095807165572, 3.83039348159519505383301947087, 4.40631948621539143839861037043, 4.75017997526825188335544832936, 5.87664044298959329900745955648, 6.42990980240923616179524075766, 6.90386867118158446352776102535, 7.36707277876973272180811318595, 7.909791602786781256767800212665, 8.043358386652882921381706620166, 8.706328387019095597966801261923, 9.077820780077449798814052905039, 9.444181284140794991678704857799, 9.741129329602803732537056492888, 10.18009326252125165046068149465, 10.32235555061267466180822521169

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