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Properties

Label 4-663552-1.1-c1e2-0-18
Degree $4$
Conductor $663552$
Sign $-1$
Analytic cond. $42.3086$
Root an. cond. $2.55039$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s + 6·17-s − 6·25-s + 6·41-s − 12·43-s − 2·49-s − 4·59-s − 4·73-s + 12·83-s + 30·89-s − 16·97-s − 20·107-s − 18·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.20·11-s + 1.45·17-s − 6/5·25-s + 0.937·41-s − 1.82·43-s − 2/7·49-s − 0.520·59-s − 0.468·73-s + 1.31·83-s + 3.17·89-s − 1.62·97-s − 1.93·107-s − 1.69·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663552 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663552 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(663552\)    =    \(2^{13} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(42.3086\)
Root analytic conductor: \(2.55039\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 663552,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ag_bq
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.29.a_o
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ag_de
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.m_di
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.e_di
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.e_g
79$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.79.a_abu
83$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.am_cg
89$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) 2.89.abe_pm
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.q_io
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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

−8.083213340314074789711735012547, −7.62787163043240924384600038350, −7.54748679580978693620147444000, −6.66594706719349938057817836550, −6.39169576392476456791762796243, −5.71142232484306515157418246668, −5.37841535852739205886879092149, −5.02019833293859749468483574463, −4.36836722296486130950332215711, −3.72360169808701325647377380441, −3.27350245786247482720696230956, −2.67267632944535564745550428540, −2.01955793349677021101374743192, −1.20016554317032358732963000682, 0, 1.20016554317032358732963000682, 2.01955793349677021101374743192, 2.67267632944535564745550428540, 3.27350245786247482720696230956, 3.72360169808701325647377380441, 4.36836722296486130950332215711, 5.02019833293859749468483574463, 5.37841535852739205886879092149, 5.71142232484306515157418246668, 6.39169576392476456791762796243, 6.66594706719349938057817836550, 7.54748679580978693620147444000, 7.62787163043240924384600038350, 8.083213340314074789711735012547

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