Properties

Label 4-663552-1.1-c1e2-0-27
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 − 2·17-s + 2·25-s − 2·41-s − 12·43-s − 10·49-s − 4·59-s − 16·67-s − 20·73-s − 12·83-s − 18·89-s − 4·107-s − 18·113-s + 6·121-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.485·17-s + 2/5·25-s − 0.312·41-s − 1.82·43-s − 1.42·49-s − 0.520·59-s − 1.95·67-s − 2.34·73-s − 1.31·83-s − 1.90·89-s − 0.386·107-s − 1.69·113-s + 6/11·121-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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.ae_k
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.c_k
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.c_de
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.m_eo
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.47.a_c
53$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.53.a_ck
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.e_di
61$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.61.a_bm
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.q_ha
71$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.71.a_adq
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.u_iw
79$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.79.a_ady
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.s_gw
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

−8.384621070850913416153815772140, −7.59975611036620556892906241169, −7.19523487874934462858002412970, −6.72917333213766140722446789448, −6.41837735856575227473623505272, −5.91364219632269512929721021497, −5.39008847706981631272183454865, −4.75898201686463271520699744061, −4.37087036507384903100010666531, −3.89745822559605834211808654810, −3.14073274970007574415170213426, −2.84715959744495461923339727678, −1.72152769979827701933766706186, −1.41614068784030805713397454471, 0, 1.41614068784030805713397454471, 1.72152769979827701933766706186, 2.84715959744495461923339727678, 3.14073274970007574415170213426, 3.89745822559605834211808654810, 4.37087036507384903100010666531, 4.75898201686463271520699744061, 5.39008847706981631272183454865, 5.91364219632269512929721021497, 6.41837735856575227473623505272, 6.72917333213766140722446789448, 7.19523487874934462858002412970, 7.59975611036620556892906241169, 8.384621070850913416153815772140

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