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Properties

Label 4-792e2-1.1-c1e2-0-21
Degree $4$
Conductor $627264$
Sign $1$
Analytic cond. $39.9948$
Root an. cond. $2.51478$
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 − 2·4-s + 9-s + 6·11-s − 2·12-s + 4·16-s − 6·17-s − 5·19-s − 7·25-s + 27-s + 6·33-s − 2·36-s + 12·41-s − 8·43-s − 12·44-s + 4·48-s − 10·49-s − 6·51-s − 5·57-s − 12·59-s − 8·64-s + 4·67-s + 12·68-s − 14·73-s − 7·75-s + 10·76-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 16-s − 1.45·17-s − 1.14·19-s − 7/5·25-s + 0.192·27-s + 1.04·33-s − 1/3·36-s + 1.87·41-s − 1.21·43-s − 1.80·44-s + 0.577·48-s − 1.42·49-s − 0.840·51-s − 0.662·57-s − 1.56·59-s − 64-s + 0.488·67-s + 1.45·68-s − 1.63·73-s − 0.808·75-s + 1.14·76-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(627264\)    =    \(2^{6} \cdot 3^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(39.9948\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 627264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.649104193\)
\(L(\frac12)\) \(\approx\) \(1.649104193\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3$C_1$ \( 1 - T \)
11$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.13.a_al
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.g_bi
19$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.f_bq
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \) 2.29.a_acb
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.31.a_abj
37$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.37.a_al
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.43.i_dp
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.53.a_ac
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.m_eo
61$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.61.a_dh
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ae_dy
71$C_2^2$ \( 1 + 97 T^{2} + p^{2} T^{4} \) 2.71.a_dt
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.83.as_je
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.as_jq
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.97.abi_sp
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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

−8.563497278700290888620524010249, −7.966484915623761323084338620176, −7.66299988950717275812294072722, −7.05400557880213236769602564027, −6.43474905653207082814548190816, −6.20978127019246724026930390945, −5.80839166685291372517676793286, −4.79804736690050296105935963580, −4.48200443293133298846670201165, −4.27373598076492906170712687702, −3.49932968965872422126062563954, −3.30713955274411311480361647815, −2.04880471576545946504161014634, −1.84489244953144007794527574189, −0.63790681358635891467654372908, 0.63790681358635891467654372908, 1.84489244953144007794527574189, 2.04880471576545946504161014634, 3.30713955274411311480361647815, 3.49932968965872422126062563954, 4.27373598076492906170712687702, 4.48200443293133298846670201165, 4.79804736690050296105935963580, 5.80839166685291372517676793286, 6.20978127019246724026930390945, 6.43474905653207082814548190816, 7.05400557880213236769602564027, 7.66299988950717275812294072722, 7.966484915623761323084338620176, 8.563497278700290888620524010249

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