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Properties

Label 4-623808-1.1-c1e2-0-33
Degree $4$
Conductor $623808$
Sign $1$
Analytic cond. $39.7745$
Root an. cond. $2.51131$
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 + 4·7-s + 9-s − 4·19-s + 4·21-s − 2·25-s + 27-s + 4·29-s + 8·41-s + 2·49-s − 8·53-s − 4·57-s + 12·59-s + 4·63-s + 12·71-s − 4·73-s − 2·75-s + 81-s + 4·87-s + 12·89-s + 4·107-s + 16·113-s + 18·121-s + 8·123-s + 127-s + 131-s − 16·133-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.917·19-s + 0.872·21-s − 2/5·25-s + 0.192·27-s + 0.742·29-s + 1.24·41-s + 2/7·49-s − 1.09·53-s − 0.529·57-s + 1.56·59-s + 0.503·63-s + 1.42·71-s − 0.468·73-s − 0.230·75-s + 1/9·81-s + 0.428·87-s + 1.27·89-s + 0.386·107-s + 1.50·113-s + 1.63·121-s + 0.721·123-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(623808\)    =    \(2^{6} \cdot 3^{3} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(39.7745\)
Root analytic conductor: \(2.51131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.002886467\)
\(L(\frac12)\) \(\approx\) \(3.002886467\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
19$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ae_o
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
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$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.ai_dq
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.47.a_de
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.i_eo
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.am_eo
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.a_di
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.67.a_dy
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.am_fm
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.e_fe
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.79.a_di
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.97.a_eg
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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

−8.359836215585878147404464295663, −8.043957134850943331707290342304, −7.60954366161212441841168959300, −7.17587924046205345837490036556, −6.61563981323718344391274043484, −6.12226441782807349326174107848, −5.63558419330464481919975959290, −4.90440703228611863908541590897, −4.71383952032959165340684047336, −4.15257835110843879444918338178, −3.62438409784530609145614783623, −2.90570333520834236949992562495, −2.18128071009031845501950732787, −1.82203951408740825792417992400, −0.883714616280201484441583169203, 0.883714616280201484441583169203, 1.82203951408740825792417992400, 2.18128071009031845501950732787, 2.90570333520834236949992562495, 3.62438409784530609145614783623, 4.15257835110843879444918338178, 4.71383952032959165340684047336, 4.90440703228611863908541590897, 5.63558419330464481919975959290, 6.12226441782807349326174107848, 6.61563981323718344391274043484, 7.17587924046205345837490036556, 7.60954366161212441841168959300, 8.043957134850943331707290342304, 8.359836215585878147404464295663

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