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Properties

Label 4-650e2-1.1-c1e2-0-30
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 4-s + 8·7-s + 8·9-s − 4·11-s − 4·12-s + 6·13-s + 16-s − 6·17-s + 4·19-s + 32·21-s − 4·23-s + 12·27-s − 8·28-s − 12·31-s − 16·33-s − 8·36-s + 8·37-s + 24·39-s − 2·41-s + 4·43-s + 4·44-s + 8·47-s + 4·48-s + 34·49-s − 24·51-s − 6·52-s + ⋯
L(s)  = 1  + 2.30·3-s − 1/2·4-s + 3.02·7-s + 8/3·9-s − 1.20·11-s − 1.15·12-s + 1.66·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 6.98·21-s − 0.834·23-s + 2.30·27-s − 1.51·28-s − 2.15·31-s − 2.78·33-s − 4/3·36-s + 1.31·37-s + 3.84·39-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + 1.16·47-s + 0.577·48-s + 34/7·49-s − 3.36·51-s − 0.832·52-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

\(L(1)\) \(\approx\) \(5.436685679\)
\(L(\frac12)\) \(\approx\) \(5.436685679\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.3.ae_i
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_i
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.g_s
19$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_i
23$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_i
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_cu
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.c_c
43$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_i
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_c
59$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_i
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.61.y_kg
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_i
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.83.q_iw
89$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.89.k_by
97$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−10.69230141346312878733575073106, −10.50995344518017094117188706889, −9.680615308790925975410767732842, −9.099932212278949182346080681223, −8.898518633300712452835778985469, −8.514638629328725393562084395657, −8.339454159776964439691924760256, −7.78130754168409653656442853786, −7.56032667219278632310416589852, −7.38307143300473749531414122598, −6.24109712071149530818496442990, −5.63754566183678985905293028138, −5.11997989042394332386228669993, −4.66359968432755867174297593366, −3.96460363608794387542605870769, −3.95602209396462671654605519498, −2.89701658083814475441146994724, −2.46480791941972673868533947489, −1.76321948263743566535957896502, −1.39431479805397634058392889079, 1.39431479805397634058392889079, 1.76321948263743566535957896502, 2.46480791941972673868533947489, 2.89701658083814475441146994724, 3.95602209396462671654605519498, 3.96460363608794387542605870769, 4.66359968432755867174297593366, 5.11997989042394332386228669993, 5.63754566183678985905293028138, 6.24109712071149530818496442990, 7.38307143300473749531414122598, 7.56032667219278632310416589852, 7.78130754168409653656442853786, 8.339454159776964439691924760256, 8.514638629328725393562084395657, 8.898518633300712452835778985469, 9.099932212278949182346080681223, 9.680615308790925975410767732842, 10.50995344518017094117188706889, 10.69230141346312878733575073106

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