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Properties

Label 4-5550e2-1.1-c1e2-0-5
Degree $4$
Conductor $30802500$
Sign $1$
Analytic cond. $1963.99$
Root an. cond. $6.65709$
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 + 3·4-s + 4·6-s + 7-s + 4·8-s + 3·9-s + 3·11-s + 6·12-s + 3·13-s + 2·14-s + 5·16-s − 3·17-s + 6·18-s + 3·19-s + 2·21-s + 6·22-s − 23-s + 8·24-s + 6·26-s + 4·27-s + 3·28-s + 12·29-s − 6·31-s + 6·32-s + 6·33-s − 6·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.377·7-s + 1.41·8-s + 9-s + 0.904·11-s + 1.73·12-s + 0.832·13-s + 0.534·14-s + 5/4·16-s − 0.727·17-s + 1.41·18-s + 0.688·19-s + 0.436·21-s + 1.27·22-s − 0.208·23-s + 1.63·24-s + 1.17·26-s + 0.769·27-s + 0.566·28-s + 2.22·29-s − 1.07·31-s + 1.06·32-s + 1.04·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30802500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(1963.99\)
Root analytic conductor: \(6.65709\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 30802500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(20.15284449\)
\(L(\frac12)\) \(\approx\) \(20.15284449\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_g
11$D_{4}$ \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_q
13$D_{4}$ \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_u
17$D_{4}$ \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_bc
19$D_{4}$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.19.ad_bg
23$D_{4}$ \( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} \) 2.23.b_bm
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_bm
41$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_cg
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$D_{4}$ \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_ck
53$D_{4}$ \( 1 + 5 T + 104 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.53.f_ea
59$D_{4}$ \( 1 - 18 T + 166 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.59.as_gk
61$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_du
67$D_{4}$ \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_eg
71$D_{4}$ \( 1 - 14 T + 158 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.71.ao_gc
73$D_{4}$ \( 1 + 9 T + 92 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.73.j_do
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$D_{4}$ \( 1 + 7 T + 170 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.83.h_go
89$D_{4}$ \( 1 - 21 T + 280 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.89.av_ku
97$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_co
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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

−8.216228123068362852020176476179, −7.957008804705988798366098842548, −7.64034471664453590170721498315, −7.14656901198569158147587521772, −6.71695877595844119838696897010, −6.67669776249369775278378642322, −6.14977434994634330761303279691, −5.87986817225805524394872683796, −5.22587016218050467252742844985, −5.08762621687267268319509311600, −4.49366390112364831504086104998, −4.25550725738654111998808176906, −3.82657155368126037494721237770, −3.61076825004540196256521852926, −3.13480480712904013430456127504, −2.70765705727071706369769103456, −2.17009591021394470659429481266, −2.02260044003993856344573316923, −1.07823350945689797624183089441, −1.01904958213642045451321757020, 1.01904958213642045451321757020, 1.07823350945689797624183089441, 2.02260044003993856344573316923, 2.17009591021394470659429481266, 2.70765705727071706369769103456, 3.13480480712904013430456127504, 3.61076825004540196256521852926, 3.82657155368126037494721237770, 4.25550725738654111998808176906, 4.49366390112364831504086104998, 5.08762621687267268319509311600, 5.22587016218050467252742844985, 5.87986817225805524394872683796, 6.14977434994634330761303279691, 6.67669776249369775278378642322, 6.71695877595844119838696897010, 7.14656901198569158147587521772, 7.64034471664453590170721498315, 7.957008804705988798366098842548, 8.216228123068362852020176476179

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