Properties

Label 4-507e2-1.1-c1e2-0-18
Degree $4$
Conductor $257049$
Sign $1$
Analytic cond. $16.3896$
Root an. cond. $2.01206$
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  + 2·2-s + 2·3-s + 4-s + 4·6-s + 3·9-s + 4·11-s + 2·12-s + 16-s + 4·17-s + 6·18-s + 8·22-s − 8·23-s − 2·25-s + 4·27-s + 4·29-s + 8·31-s − 2·32-s + 8·33-s + 8·34-s + 3·36-s + 4·37-s − 16·41-s + 8·43-s + 4·44-s − 16·46-s + 12·47-s + 2·48-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s + 9-s + 1.20·11-s + 0.577·12-s + 1/4·16-s + 0.970·17-s + 1.41·18-s + 1.70·22-s − 1.66·23-s − 2/5·25-s + 0.769·27-s + 0.742·29-s + 1.43·31-s − 0.353·32-s + 1.39·33-s + 1.37·34-s + 1/2·36-s + 0.657·37-s − 2.49·41-s + 1.21·43-s + 0.603·44-s − 2.35·46-s + 1.75·47-s + 0.288·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(257049\)    =    \(3^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(16.3896\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{507} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 257049,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.759994761\)
\(L(\frac12)\) \(\approx\) \(5.759994761\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−11.40636187453181820218716400890, −10.46353179355012938863685529446, −10.29009474604155982861718261685, −9.776573267697640968952269992143, −9.440993227757815417901500224992, −8.804453106396835433251662182094, −8.437343651733120170878274050456, −8.043318461451174999840626860467, −7.42810799455639516944209003717, −7.13234674191477123771960679152, −6.17980760568330452251597112081, −6.17544206650059337330794595692, −5.42525868533108854512013383441, −4.70363284754302942564370984901, −4.30286503434185049344072381883, −3.99049135170150732322088398413, −3.36008278962898975441928512649, −2.94405368280544058026564270516, −2.02541190877468342186149530379, −1.28541537059419729366792066944, 1.28541537059419729366792066944, 2.02541190877468342186149530379, 2.94405368280544058026564270516, 3.36008278962898975441928512649, 3.99049135170150732322088398413, 4.30286503434185049344072381883, 4.70363284754302942564370984901, 5.42525868533108854512013383441, 6.17544206650059337330794595692, 6.17980760568330452251597112081, 7.13234674191477123771960679152, 7.42810799455639516944209003717, 8.043318461451174999840626860467, 8.437343651733120170878274050456, 8.804453106396835433251662182094, 9.440993227757815417901500224992, 9.776573267697640968952269992143, 10.29009474604155982861718261685, 10.46353179355012938863685529446, 11.40636187453181820218716400890

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