Properties

Label 4-415872-1.1-c1e2-0-33
Degree $4$
Conductor $415872$
Sign $1$
Analytic cond. $26.5163$
Root an. cond. $2.26922$
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  + 2-s + 3·3-s + 4-s + 3·6-s + 8-s + 6·9-s + 3·12-s + 16-s + 6·18-s + 8·19-s + 3·24-s + 2·25-s + 9·27-s + 32-s + 6·36-s + 8·38-s − 4·43-s + 3·48-s − 13·49-s + 2·50-s + 9·54-s + 24·57-s − 6·59-s + 64-s + 6·72-s − 22·73-s + 6·75-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.22·6-s + 0.353·8-s + 2·9-s + 0.866·12-s + 1/4·16-s + 1.41·18-s + 1.83·19-s + 0.612·24-s + 2/5·25-s + 1.73·27-s + 0.176·32-s + 36-s + 1.29·38-s − 0.609·43-s + 0.433·48-s − 1.85·49-s + 0.282·50-s + 1.22·54-s + 3.17·57-s − 0.781·59-s + 1/8·64-s + 0.707·72-s − 2.57·73-s + 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(415872\)    =    \(2^{7} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(26.5163\)
Root analytic conductor: \(2.26922\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 415872,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.139818101\)
\(L(\frac12)\) \(\approx\) \(6.139818101\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_2$ \( 1 - p T + p T^{2} \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.13.a_x
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.17.a_abf
23$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.23.a_t
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.43.e_dm
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.47.a_de
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.59.g_ex
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \) 2.67.a_ach
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.79.a_eg
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$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 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

−8.650006878665126244618813702581, −8.134822752393194998575286471882, −7.62904025825301046186028084804, −7.38891725579909033971910904855, −6.97049407956336384207438784156, −6.29468661547540970579688800446, −5.85532409349730578043944208089, −5.03978044028836672994308230391, −4.78938031771961950710214108675, −4.13862501481106319527729444195, −3.42171154743551676241609849178, −3.21870034436397941270835984622, −2.70678198949014858718396855564, −1.89788314102936140312185908888, −1.27315653782020057953763083056, 1.27315653782020057953763083056, 1.89788314102936140312185908888, 2.70678198949014858718396855564, 3.21870034436397941270835984622, 3.42171154743551676241609849178, 4.13862501481106319527729444195, 4.78938031771961950710214108675, 5.03978044028836672994308230391, 5.85532409349730578043944208089, 6.29468661547540970579688800446, 6.97049407956336384207438784156, 7.38891725579909033971910904855, 7.62904025825301046186028084804, 8.134822752393194998575286471882, 8.650006878665126244618813702581

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