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Properties

Label 4-650e2-1.1-c1e2-0-18
Degree $4$
Conductor $422500$
Sign $-1$
Analytic cond. $26.9389$
Root an. cond. $2.27821$
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 − 2·7-s + 3·8-s − 4·9-s + 2·13-s + 2·14-s − 16-s + 4·18-s − 2·26-s + 2·28-s + 8·29-s − 5·32-s + 4·36-s − 8·37-s − 2·47-s − 10·49-s − 2·52-s − 6·56-s − 8·58-s + 8·63-s + 7·64-s + 22·67-s − 12·72-s − 4·73-s + 8·74-s + 7·81-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s − 4/3·9-s + 0.554·13-s + 0.534·14-s − 1/4·16-s + 0.942·18-s − 0.392·26-s + 0.377·28-s + 1.48·29-s − 0.883·32-s + 2/3·36-s − 1.31·37-s − 0.291·47-s − 1.42·49-s − 0.277·52-s − 0.801·56-s − 1.05·58-s + 1.00·63-s + 7/8·64-s + 2.68·67-s − 1.41·72-s − 0.468·73-s + 0.929·74-s + 7/9·81-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: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 422500,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.c_o
11$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.11.a_m
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.19.a_ae
23$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.23.a_au
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ai_cg
31$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.31.a_au
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.i_cc
41$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.41.a_cg
43$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.43.a_e
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.c_dq
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.53.a_k
59$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.59.a_ae
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.67.aw_ju
71$C_2^2$ \( 1 + 92 T^{2} + p^{2} T^{4} \) 2.71.a_do
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.83.ak_fm
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.89.a_afa
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.aq_io
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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

−8.502148767745462064307435277978, −8.195608794202196063992207406336, −7.64281219886484213770916841228, −6.99517732346350733966524142977, −6.55579030262102705885825381949, −6.15610820083616662019819838937, −5.58364198280950482351523022626, −5.00194093461418000275326246636, −4.69645527741399380369062909292, −3.74207331567753203247718779207, −3.45933053356887752025656521234, −2.79804707834057758971541082257, −2.00676274304805294037655770739, −0.979266316456152354006782295374, 0, 0.979266316456152354006782295374, 2.00676274304805294037655770739, 2.79804707834057758971541082257, 3.45933053356887752025656521234, 3.74207331567753203247718779207, 4.69645527741399380369062909292, 5.00194093461418000275326246636, 5.58364198280950482351523022626, 6.15610820083616662019819838937, 6.55579030262102705885825381949, 6.99517732346350733966524142977, 7.64281219886484213770916841228, 8.195608794202196063992207406336, 8.502148767745462064307435277978

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