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Properties

Label 4-3675e2-1.1-c1e2-0-1
Degree $4$
Conductor $13505625$
Sign $1$
Analytic cond. $861.130$
Root an. cond. $5.41710$
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  − 2·2-s − 2·3-s + 4-s + 4·6-s + 3·9-s − 2·12-s + 4·13-s + 16-s + 4·17-s − 6·18-s + 4·19-s + 4·23-s − 8·26-s − 4·27-s + 12·29-s − 4·31-s + 2·32-s − 8·34-s + 3·36-s − 4·37-s − 8·38-s − 8·39-s + 12·41-s − 16·43-s − 8·46-s − 2·48-s − 8·51-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s + 9-s − 0.577·12-s + 1.10·13-s + 1/4·16-s + 0.970·17-s − 1.41·18-s + 0.917·19-s + 0.834·23-s − 1.56·26-s − 0.769·27-s + 2.22·29-s − 0.718·31-s + 0.353·32-s − 1.37·34-s + 1/2·36-s − 0.657·37-s − 1.29·38-s − 1.28·39-s + 1.87·41-s − 2.43·43-s − 1.17·46-s − 0.288·48-s − 1.12·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13505625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(861.130\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13505625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7950647559\)
\(L(\frac12)\) \(\approx\) \(0.7950647559\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
7 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_d
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_w
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_g
19$D_{4}$ \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_bi
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_s
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_4$ \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_ag
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_bu
41$C_4$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_di
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.47.a_ck
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.53.a_bi
59$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.59.aq_fu
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.71.a_fe
73$D_{4}$ \( 1 - 28 T + 334 T^{2} - 28 p T^{3} + p^{2} T^{4} \) 2.73.abc_mw
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$D_{4}$ \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_fu
89$C_4$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_di
97$D_{4}$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_ew
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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

−8.571327948235967976946009501396, −8.559944857232549410786859797932, −7.995283391062523981352608948989, −7.79254177442945549336412054769, −7.17626620246067677294004355759, −7.02578906095780009756045214460, −6.47978991590982584859260278994, −6.13639690037221002717290866485, −5.95410471371083595305361060187, −5.24538762103696800398330916570, −5.00145870873983000882913422322, −4.85949880930655039846534616692, −3.98768933666689880663247217387, −3.70609481425852835797478177266, −3.21105630717060706787434498900, −2.72374559567764598901227615748, −1.97670019864406969303353388169, −1.17408263781867680543618585404, −1.02882918290376231323759197309, −0.50859873973369279042624339557, 0.50859873973369279042624339557, 1.02882918290376231323759197309, 1.17408263781867680543618585404, 1.97670019864406969303353388169, 2.72374559567764598901227615748, 3.21105630717060706787434498900, 3.70609481425852835797478177266, 3.98768933666689880663247217387, 4.85949880930655039846534616692, 5.00145870873983000882913422322, 5.24538762103696800398330916570, 5.95410471371083595305361060187, 6.13639690037221002717290866485, 6.47978991590982584859260278994, 7.02578906095780009756045214460, 7.17626620246067677294004355759, 7.79254177442945549336412054769, 7.995283391062523981352608948989, 8.559944857232549410786859797932, 8.571327948235967976946009501396

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