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Properties

Label 4-416000-1.1-c1e2-0-8
Degree $4$
Conductor $416000$
Sign $1$
Analytic cond. $26.5245$
Root an. cond. $2.26940$
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  + 5-s + 9-s − 5·13-s + 9·17-s + 25-s − 29-s + 5·37-s + 5·41-s + 45-s + 11·49-s + 2·53-s − 8·61-s − 5·65-s + 10·73-s − 8·81-s + 9·85-s + 6·89-s − 10·97-s + 101-s − 14·109-s − 7·113-s − 5·117-s + 121-s + 125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.447·5-s + 1/3·9-s − 1.38·13-s + 2.18·17-s + 1/5·25-s − 0.185·29-s + 0.821·37-s + 0.780·41-s + 0.149·45-s + 11/7·49-s + 0.274·53-s − 1.02·61-s − 0.620·65-s + 1.17·73-s − 8/9·81-s + 0.976·85-s + 0.635·89-s − 1.01·97-s + 0.0995·101-s − 1.34·109-s − 0.658·113-s − 0.462·117-s + 1/11·121-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(416000\)    =    \(2^{8} \cdot 5^{3} \cdot 13\)
Sign: $1$
Analytic conductor: \(26.5245\)
Root analytic conductor: \(2.26940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 416000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.101238683\)
\(L(\frac12)\) \(\approx\) \(2.101238683\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( 1 - T \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.a_al
11$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.11.a_ab
17$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.17.aj_cc
19$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.19.a_v
23$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.23.a_u
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) 2.29.b_cg
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.af_ci
41$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.af_q
43$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.43.a_ab
47$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.47.a_acn
53$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.ac_h
59$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.59.a_z
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.67.a_acc
71$C_2^2$ \( 1 - 84 T^{2} + p^{2} T^{4} \) 2.71.a_adg
73$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.ak_es
79$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \) 2.79.a_adl
83$C_2^2$ \( 1 - 88 T^{2} + p^{2} T^{4} \) 2.83.a_adk
89$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.89.ag_he
97$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.k_ih
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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

−8.594425376712416073748455627321, −8.028259693517611808804469074668, −7.72915443960381544261275474549, −7.22734861700800275214027552035, −6.98978900034914658172441744462, −6.15185570876512809251805779305, −5.82324707034002315192167994652, −5.30490162070476470113070711334, −4.93316208084899694270077958004, −4.23502642849489224121489836131, −3.74278319843780399170759650647, −2.93392304033405161432901697857, −2.55552454832398774836968455952, −1.68384043083618349318245015079, −0.831871704257557466309219548527, 0.831871704257557466309219548527, 1.68384043083618349318245015079, 2.55552454832398774836968455952, 2.93392304033405161432901697857, 3.74278319843780399170759650647, 4.23502642849489224121489836131, 4.93316208084899694270077958004, 5.30490162070476470113070711334, 5.82324707034002315192167994652, 6.15185570876512809251805779305, 6.98978900034914658172441744462, 7.22734861700800275214027552035, 7.72915443960381544261275474549, 8.028259693517611808804469074668, 8.594425376712416073748455627321

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