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Properties

Label 4-368e2-1.1-c1e2-0-9
Degree $4$
Conductor $135424$
Sign $1$
Analytic cond. $8.63475$
Root an. cond. $1.71420$
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 + 2·4-s + 4·6-s + 2·9-s + 8·11-s + 4·12-s − 6·13-s − 4·16-s − 4·17-s + 4·18-s + 16·22-s − 12·26-s + 6·27-s + 14·29-s + 4·31-s − 8·32-s + 16·33-s − 8·34-s + 4·36-s + 8·37-s − 12·39-s − 8·43-s + 16·44-s − 8·48-s − 2·49-s − 8·51-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 2/3·9-s + 2.41·11-s + 1.15·12-s − 1.66·13-s − 16-s − 0.970·17-s + 0.942·18-s + 3.41·22-s − 2.35·26-s + 1.15·27-s + 2.59·29-s + 0.718·31-s − 1.41·32-s + 2.78·33-s − 1.37·34-s + 2/3·36-s + 1.31·37-s − 1.92·39-s − 1.21·43-s + 2.41·44-s − 1.15·48-s − 2/7·49-s − 1.12·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(135424\)    =    \(2^{8} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(8.63475\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 135424,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.075262385\)
\(L(\frac12)\) \(\approx\) \(5.075262385\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
5$C_2^2$ \( 1 + p^{2} T^{4} \) 2.5.a_a
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.11.ai_bg
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_s
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.29.ao_du
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_bg
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_bg
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.53.q_ey
59$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_du
61$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_i
67$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_bg
71$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.71.a_cc
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.83.ai_bg
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−12.06405003571749166823012809707, −11.36387821674999569325546176916, −10.95558083309029648444831939764, −10.07198832115740968967108547663, −9.742873861262560381185321382052, −9.336511776146398879549014842424, −8.914501515420554901772898283376, −8.272205048388940996844110924949, −8.141460975365559579415243351109, −7.01431035498464557021768319641, −6.68726371160253003477760300878, −6.63070554317015348478838123544, −5.86070609779474056281393104011, −4.88002365486117064789566382443, −4.57914800205004704045163082110, −4.22671008582789310146663312296, −3.53314278490370372718784437860, −2.75923361975828761520560075186, −2.55739915143767689554622403566, −1.39467549267263300822400929644, 1.39467549267263300822400929644, 2.55739915143767689554622403566, 2.75923361975828761520560075186, 3.53314278490370372718784437860, 4.22671008582789310146663312296, 4.57914800205004704045163082110, 4.88002365486117064789566382443, 5.86070609779474056281393104011, 6.63070554317015348478838123544, 6.68726371160253003477760300878, 7.01431035498464557021768319641, 8.141460975365559579415243351109, 8.272205048388940996844110924949, 8.914501515420554901772898283376, 9.336511776146398879549014842424, 9.742873861262560381185321382052, 10.07198832115740968967108547663, 10.95558083309029648444831939764, 11.36387821674999569325546176916, 12.06405003571749166823012809707

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