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Properties

Label 4-245e2-1.1-c1e2-0-5
Degree $4$
Conductor $60025$
Sign $1$
Analytic cond. $3.82724$
Root an. cond. $1.39869$
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  + 3-s + 2·4-s − 5-s + 3·9-s + 3·11-s + 2·12-s − 10·13-s − 15-s + 3·17-s + 2·19-s − 2·20-s + 6·23-s + 8·27-s + 6·29-s − 4·31-s + 3·33-s + 6·36-s − 2·37-s − 10·39-s + 24·41-s − 20·43-s + 6·44-s − 3·45-s + 9·47-s + 3·51-s − 20·52-s − 12·53-s + ⋯
L(s)  = 1  + 0.577·3-s + 4-s − 0.447·5-s + 9-s + 0.904·11-s + 0.577·12-s − 2.77·13-s − 0.258·15-s + 0.727·17-s + 0.458·19-s − 0.447·20-s + 1.25·23-s + 1.53·27-s + 1.11·29-s − 0.718·31-s + 0.522·33-s + 36-s − 0.328·37-s − 1.60·39-s + 3.74·41-s − 3.04·43-s + 0.904·44-s − 0.447·45-s + 1.31·47-s + 0.420·51-s − 2.77·52-s − 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(60025\)    =    \(5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.82724\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 60025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.083808682\)
\(L(\frac12)\) \(\approx\) \(2.083808682\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.13.k_bz
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_ap
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.41.ay_is
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_bi
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_dn
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_d
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_acr
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_cd
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.97.ac_hn
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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

−12.32392450593324665056395344160, −11.88970189943156110422580026500, −11.46080155353028838978960464645, −11.00288659065086848832981884739, −10.28556129907529394333505686566, −9.952359872216849956089022052653, −9.484976221237438958588851238989, −9.084317711588151097027953548515, −8.352192050458797959516434547587, −7.70998033654721587309174344678, −7.24233835928023693117242248699, −7.07274899484813142048842616642, −6.61123773043983510273189327447, −5.71690850778224111057713378588, −4.74644062665126137294936604230, −4.69368961275903088970943567662, −3.67353999802657571057822330118, −2.81610322323524882372353158831, −2.46794845066265842749121387058, −1.30773830699806101421822577632, 1.30773830699806101421822577632, 2.46794845066265842749121387058, 2.81610322323524882372353158831, 3.67353999802657571057822330118, 4.69368961275903088970943567662, 4.74644062665126137294936604230, 5.71690850778224111057713378588, 6.61123773043983510273189327447, 7.07274899484813142048842616642, 7.24233835928023693117242248699, 7.70998033654721587309174344678, 8.352192050458797959516434547587, 9.084317711588151097027953548515, 9.484976221237438958588851238989, 9.952359872216849956089022052653, 10.28556129907529394333505686566, 11.00288659065086848832981884739, 11.46080155353028838978960464645, 11.88970189943156110422580026500, 12.32392450593324665056395344160

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