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Properties

Label 4-4950e2-1.1-c1e2-0-37
Degree $4$
Conductor $24502500$
Sign $1$
Analytic cond. $1562.30$
Root an. cond. $6.28696$
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  − 2·2-s + 3·4-s − 7-s − 4·8-s + 2·11-s − 4·13-s + 2·14-s + 5·16-s − 3·17-s + 7·19-s − 4·22-s − 6·23-s + 8·26-s − 3·28-s + 3·29-s + 31-s − 6·32-s + 6·34-s − 13·37-s − 14·38-s + 8·43-s + 6·44-s + 12·46-s − 6·47-s − 5·49-s − 12·52-s + 9·53-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s + 0.603·11-s − 1.10·13-s + 0.534·14-s + 5/4·16-s − 0.727·17-s + 1.60·19-s − 0.852·22-s − 1.25·23-s + 1.56·26-s − 0.566·28-s + 0.557·29-s + 0.179·31-s − 1.06·32-s + 1.02·34-s − 2.13·37-s − 2.27·38-s + 1.21·43-s + 0.904·44-s + 1.76·46-s − 0.875·47-s − 5/7·49-s − 1.66·52-s + 1.23·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24502500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1562.30\)
Root analytic conductor: \(6.28696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 24502500,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_g
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$D_{4}$ \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_bc
19$D_{4}$ \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.19.ah_bq
23$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_w
29$D_{4}$ \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_ca
31$D_{4}$ \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \) 2.31.ab_cc
37$D_{4}$ \( 1 + 13 T + 108 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.37.n_ee
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.41.a_aby
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$D_{4}$ \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_cs
53$D_{4}$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_ca
59$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_dq
61$D_{4}$ \( 1 + 5 T + 120 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_eq
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$D_{4}$ \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.71.d_cs
73$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_be
79$D_{4}$ \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_gs
83$D_{4}$ \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_fm
89$D_{4}$ \( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.89.d_gq
97$D_{4}$ \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.97.ao_ic
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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

−8.017637585006348520227328644935, −7.79294763549993325217619808678, −7.27710101114682377490844291281, −7.26103110883045301954338705360, −6.66380799495427026102804544071, −6.53832530644259488053929343841, −5.95602454770431659440922950217, −5.74760645779859951798545448350, −5.05554384694337290805841813534, −4.98441891198832452769047333050, −4.19712402740535266970959542782, −3.97151016297051327878671560357, −3.16879138609288381824344005953, −3.14533124261814314153039276429, −2.43455653025072204342386265506, −2.12449234460290937536210726977, −1.41439500488856941593153725874, −1.16253671815266753101452741456, 0, 0, 1.16253671815266753101452741456, 1.41439500488856941593153725874, 2.12449234460290937536210726977, 2.43455653025072204342386265506, 3.14533124261814314153039276429, 3.16879138609288381824344005953, 3.97151016297051327878671560357, 4.19712402740535266970959542782, 4.98441891198832452769047333050, 5.05554384694337290805841813534, 5.74760645779859951798545448350, 5.95602454770431659440922950217, 6.53832530644259488053929343841, 6.66380799495427026102804544071, 7.26103110883045301954338705360, 7.27710101114682377490844291281, 7.79294763549993325217619808678, 8.017637585006348520227328644935

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