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Properties

Label 4-650e2-1.1-c1e2-0-3
Degree $4$
Conductor $422500$
Sign $1$
Analytic cond. $26.9389$
Root an. cond. $2.27821$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 4·7-s + 2·9-s − 2·11-s + 2·12-s − 4·13-s + 16-s + 2·17-s − 6·19-s − 8·21-s − 2·23-s − 6·27-s − 4·28-s + 2·31-s + 4·33-s − 2·36-s − 16·37-s + 8·39-s − 14·41-s − 2·43-s + 2·44-s + 20·47-s − 2·48-s − 2·49-s − 4·51-s + 4·52-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 1.51·7-s + 2/3·9-s − 0.603·11-s + 0.577·12-s − 1.10·13-s + 1/4·16-s + 0.485·17-s − 1.37·19-s − 1.74·21-s − 0.417·23-s − 1.15·27-s − 0.755·28-s + 0.359·31-s + 0.696·33-s − 1/3·36-s − 2.63·37-s + 1.28·39-s − 2.18·41-s − 0.304·43-s + 0.301·44-s + 2.91·47-s − 0.288·48-s − 2/7·49-s − 0.560·51-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(422500\)    =    \(2^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(26.9389\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6420711061\)
\(L(\frac12)\) \(\approx\) \(0.6420711061\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_c
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_c
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_c
19$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_s
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_c
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_c
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.41.o_du
43$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_c
47$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.47.au_hm
53$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_c
59$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.59.s_gg
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.71.ak_by
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.83.abk_sw
89$C_2^2$ \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.89.aw_ji
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.97.a_c
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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

−10.73303721600439363760297218680, −10.34325406579499916026999705077, −10.29361584125900938126828197301, −9.365453398881506248531166959616, −9.201596135118476114842273748172, −8.324908167818270776903304483087, −8.306710781180406186907575547740, −7.59393914082725413081402828852, −7.45007088155384587541795366382, −6.56709210887130089878145894380, −6.40158079755544011129432915351, −5.48047257650686113711776995125, −5.30055207529574521089037101435, −4.91395577804579494066940500857, −4.53586791952434054791300326720, −3.86546563358439182090976397331, −3.20614169614609321363323254999, −1.96998103694287451206770222518, −1.88824070217335622871656990854, −0.45910952408094615663762482309, 0.45910952408094615663762482309, 1.88824070217335622871656990854, 1.96998103694287451206770222518, 3.20614169614609321363323254999, 3.86546563358439182090976397331, 4.53586791952434054791300326720, 4.91395577804579494066940500857, 5.30055207529574521089037101435, 5.48047257650686113711776995125, 6.40158079755544011129432915351, 6.56709210887130089878145894380, 7.45007088155384587541795366382, 7.59393914082725413081402828852, 8.306710781180406186907575547740, 8.324908167818270776903304483087, 9.201596135118476114842273748172, 9.365453398881506248531166959616, 10.29361584125900938126828197301, 10.34325406579499916026999705077, 10.73303721600439363760297218680

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