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Properties

Label 4-864e2-1.1-c1e2-0-9
Degree $4$
Conductor $746496$
Sign $1$
Analytic cond. $47.5972$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 6·11-s + 6·17-s + 8·19-s − 7·25-s − 27-s − 6·33-s − 6·41-s + 2·43-s − 49-s − 6·51-s − 8·57-s + 12·59-s + 17·67-s − 5·73-s + 7·75-s + 81-s − 12·89-s + 22·97-s + 6·99-s − 30·107-s + 14·121-s + 6·123-s + 127-s − 2·129-s + 131-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.80·11-s + 1.45·17-s + 1.83·19-s − 7/5·25-s − 0.192·27-s − 1.04·33-s − 0.937·41-s + 0.304·43-s − 1/7·49-s − 0.840·51-s − 1.05·57-s + 1.56·59-s + 2.07·67-s − 0.585·73-s + 0.808·75-s + 1/9·81-s − 1.27·89-s + 2.23·97-s + 0.603·99-s − 2.90·107-s + 1.27·121-s + 0.541·123-s + 0.0887·127-s − 0.176·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(746496\)    =    \(2^{10} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(47.5972\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 746496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.126418235\)
\(L(\frac12)\) \(\approx\) \(2.126418235\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.7.a_b
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ag_w
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ag_bi
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.ai_bt
23$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.23.a_al
29$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.29.a_t
31$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.31.a_b
37$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.37.a_n
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.g_de
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.43.ac_dj
47$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.47.a_abj
53$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \) 2.53.a_acz
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.61.a_al
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.ar_he
71$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.71.a_dh
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.f_fc
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.79.a_fa
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.m_gw
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.97.aw_kt
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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

−8.133949766638015311131647167836, −7.905689899768977012270748643488, −7.26359254217454229076445169252, −6.93025977408625566618503554868, −6.58754514394795215694997382243, −5.87558128376575952806999392759, −5.59860662553193168857255578291, −5.27460294001417176784611082179, −4.54286766179369215026083020795, −3.99154168047157592477516456010, −3.52821439292886582676068764539, −3.18251050781762512246725363966, −2.12362495002012590435855573626, −1.38837448636201565825473352073, −0.845698580927004502316312392262, 0.845698580927004502316312392262, 1.38837448636201565825473352073, 2.12362495002012590435855573626, 3.18251050781762512246725363966, 3.52821439292886582676068764539, 3.99154168047157592477516456010, 4.54286766179369215026083020795, 5.27460294001417176784611082179, 5.59860662553193168857255578291, 5.87558128376575952806999392759, 6.58754514394795215694997382243, 6.93025977408625566618503554868, 7.26359254217454229076445169252, 7.905689899768977012270748643488, 8.133949766638015311131647167836

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