Properties

Label 4-260000-1.1-c1e2-0-11
Degree $4$
Conductor $260000$
Sign $-1$
Analytic cond. $16.5778$
Root an. cond. $2.01781$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 9-s − 7·13-s + 16-s + 17-s − 18-s − 7·26-s − 29-s + 32-s + 34-s − 36-s − 9·37-s − 5·41-s − 11·49-s − 7·52-s − 20·53-s − 58-s + 8·61-s + 64-s + 68-s − 72-s − 14·73-s − 9·74-s − 8·81-s − 5·82-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1/3·9-s − 1.94·13-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.37·26-s − 0.185·29-s + 0.176·32-s + 0.171·34-s − 1/6·36-s − 1.47·37-s − 0.780·41-s − 1.57·49-s − 0.970·52-s − 2.74·53-s − 0.131·58-s + 1.02·61-s + 1/8·64-s + 0.121·68-s − 0.117·72-s − 1.63·73-s − 1.04·74-s − 8/9·81-s − 0.552·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(260000\)    =    \(2^{5} \cdot 5^{4} \cdot 13\)
Sign: $-1$
Analytic conductor: \(16.5778\)
Root analytic conductor: \(2.01781\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 260000,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5 \( 1 \)
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_b
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
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.ab_o
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_au
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$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.j_dk
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.41.f_q
43$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.43.a_b
47$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \) 2.47.a_cn
53$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.53.u_hx
59$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.59.a_z
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ai_dy
67$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.67.a_cc
71$C_2^2$ \( 1 - 84 T^{2} + p^{2} T^{4} \) 2.71.a_adg
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.73.o_go
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_dk
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$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.97.ae_gr
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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

−8.571935241062024994204253747977, −8.182240000858389151505766254501, −7.61815719270988739806765557077, −7.22444792492923758143107618245, −6.80681757507756034193362573985, −6.25643725040790029362171222592, −5.69562541474937615762650066515, −5.12244141825260769960287200644, −4.79759873792514995724297749989, −4.34115469280767342267181026869, −3.28693016144256894418196653146, −3.19482101845100822137332901705, −2.26606041702379055650695462273, −1.66088958983705807039046217863, 0, 1.66088958983705807039046217863, 2.26606041702379055650695462273, 3.19482101845100822137332901705, 3.28693016144256894418196653146, 4.34115469280767342267181026869, 4.79759873792514995724297749989, 5.12244141825260769960287200644, 5.69562541474937615762650066515, 6.25643725040790029362171222592, 6.80681757507756034193362573985, 7.22444792492923758143107618245, 7.61815719270988739806765557077, 8.182240000858389151505766254501, 8.571935241062024994204253747977

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