Properties

Label 4-676000-1.1-c1e2-0-28
Degree $4$
Conductor $676000$
Sign $-1$
Analytic cond. $43.1023$
Root an. cond. $2.56227$
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 − 5-s − 8-s + 5·9-s + 10-s − 3·13-s + 16-s − 7·17-s − 5·18-s − 20-s + 25-s + 3·26-s + 5·29-s − 32-s + 7·34-s + 5·36-s + 6·37-s + 40-s + 10·41-s − 5·45-s − 8·49-s − 50-s − 3·52-s − 4·53-s − 5·58-s − 7·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 5/3·9-s + 0.316·10-s − 0.832·13-s + 1/4·16-s − 1.69·17-s − 1.17·18-s − 0.223·20-s + 1/5·25-s + 0.588·26-s + 0.928·29-s − 0.176·32-s + 1.20·34-s + 5/6·36-s + 0.986·37-s + 0.158·40-s + 1.56·41-s − 0.745·45-s − 8/7·49-s − 0.141·50-s − 0.416·52-s − 0.549·53-s − 0.656·58-s − 0.896·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(676000\)    =    \(2^{5} \cdot 5^{3} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(43.1023\)
Root analytic conductor: \(2.56227\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
13$C_2$ \( 1 + 3 T + p T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.7.a_i
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.11.a_u
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.17.h_bs
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
23$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.23.a_e
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.29.af_ca
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.31.a_abl
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ag_cg
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.41.ak_ec
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.47.a_bs
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.53.e_al
59$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.59.a_cm
61$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.h_ek
67$C_2^2$ \( 1 + 129 T^{2} + p^{2} T^{4} \) 2.67.a_ez
71$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.71.a_dp
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.73.t_iq
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.79.a_ac
83$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \) 2.83.a_aev
89$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.y_lu
97$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.q_ju
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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

−8.061894665364060011452661412880, −7.70874056423425564199205049247, −7.24256877209522805465705180206, −6.82326795619448703100508651431, −6.62010192933536321881515275041, −5.99962003015719845795492612993, −5.35757129780083269223911288193, −4.56473997449825872102778252984, −4.38466096916273616818195439858, −4.07651928918244452924110956573, −2.99779683415802512249875957384, −2.62791808527079300680577529803, −1.82202153507372323442418676583, −1.17733220359895748462855568233, 0, 1.17733220359895748462855568233, 1.82202153507372323442418676583, 2.62791808527079300680577529803, 2.99779683415802512249875957384, 4.07651928918244452924110956573, 4.38466096916273616818195439858, 4.56473997449825872102778252984, 5.35757129780083269223911288193, 5.99962003015719845795492612993, 6.62010192933536321881515275041, 6.82326795619448703100508651431, 7.24256877209522805465705180206, 7.70874056423425564199205049247, 8.061894665364060011452661412880

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