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Properties

Label 4-155e2-1.1-c1e2-0-2
Degree $4$
Conductor $24025$
Sign $1$
Analytic cond. $1.53185$
Root an. cond. $1.11251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 2·3-s + 8·4-s − 5-s + 8·6-s − 4·7-s − 8·8-s + 3·9-s + 4·10-s − 5·11-s − 16·12-s − 6·13-s + 16·14-s + 2·15-s − 4·16-s + 4·17-s − 12·18-s − 4·19-s − 8·20-s + 8·21-s + 20·22-s − 4·23-s + 16·24-s + 24·26-s − 10·27-s − 32·28-s − 6·29-s + ⋯
L(s)  = 1  − 2.82·2-s − 1.15·3-s + 4·4-s − 0.447·5-s + 3.26·6-s − 1.51·7-s − 2.82·8-s + 9-s + 1.26·10-s − 1.50·11-s − 4.61·12-s − 1.66·13-s + 4.27·14-s + 0.516·15-s − 16-s + 0.970·17-s − 2.82·18-s − 0.917·19-s − 1.78·20-s + 1.74·21-s + 4.26·22-s − 0.834·23-s + 3.26·24-s + 4.70·26-s − 1.92·27-s − 6.04·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24025\)    =    \(5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(1.53185\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 24025,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad5$C_2$ \( 1 + T + T^{2} \)
31$C_2$ \( 1 - 11 T + p T^{2} \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \) 2.2.e_i
3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.e_j
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_o
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_ab
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_ad
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.23.e_by
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bb
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.43.a_abr
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.59.n_eg
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.61.ac_et
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2^2$ \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.71.d_ack
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_acf
79$C_2^2$ \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.79.d_acs
83$C_2^2$ \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_acp
89$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.89.w_ln
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−12.24429313981002902893675049744, −12.23102215932292454761592692382, −11.22659064179652138012995802571, −11.05922806610700767277356309305, −10.16239271886388953826871307646, −10.16118483424054760471432519820, −9.738242727776993384435802923301, −9.466941140208325930717422566762, −8.558353043680014360499139782255, −7.84321205027008286735775699455, −7.80933866778182779189857014811, −7.11145952898096031435857654309, −6.64169641297630884588931302935, −5.92875796620519153410617600255, −5.05330981387258568658375320057, −4.34848311770244412723584292143, −2.96408161099442873083837533697, −1.88919001711043863429332122796, 0, 0, 1.88919001711043863429332122796, 2.96408161099442873083837533697, 4.34848311770244412723584292143, 5.05330981387258568658375320057, 5.92875796620519153410617600255, 6.64169641297630884588931302935, 7.11145952898096031435857654309, 7.80933866778182779189857014811, 7.84321205027008286735775699455, 8.558353043680014360499139782255, 9.466941140208325930717422566762, 9.738242727776993384435802923301, 10.16118483424054760471432519820, 10.16239271886388953826871307646, 11.05922806610700767277356309305, 11.22659064179652138012995802571, 12.23102215932292454761592692382, 12.24429313981002902893675049744

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