Properties

Label 4-5310e2-1.1-c1e2-0-11
Degree $4$
Conductor $28196100$
Sign $1$
Analytic cond. $1797.80$
Root an. cond. $6.51157$
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 − 2·5-s + 2·7-s + 4·8-s − 4·10-s − 2·11-s − 8·13-s + 4·14-s + 5·16-s − 12·17-s + 8·19-s − 6·20-s − 4·22-s + 12·23-s + 3·25-s − 16·26-s + 6·28-s − 8·29-s + 6·32-s − 24·34-s − 4·35-s − 8·37-s + 16·38-s − 8·40-s − 8·41-s − 8·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s + 1.41·8-s − 1.26·10-s − 0.603·11-s − 2.21·13-s + 1.06·14-s + 5/4·16-s − 2.91·17-s + 1.83·19-s − 1.34·20-s − 0.852·22-s + 2.50·23-s + 3/5·25-s − 3.13·26-s + 1.13·28-s − 1.48·29-s + 1.06·32-s − 4.11·34-s − 0.676·35-s − 1.31·37-s + 2.59·38-s − 1.26·40-s − 1.24·41-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28196100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28196100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28196100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 59^{2}\)
Sign: $1$
Analytic conductor: \(1797.80\)
Root analytic conductor: \(6.51157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 28196100,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
59$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_d
11$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_l
13$D_{4}$ \( 1 + 8 T + 3 p T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bn
17$D_{4}$ \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.17.m_cp
19$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.19.ai_bq
23$D_{4}$ \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.23.am_cs
29$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_ck
31$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.31.a_by
37$D_{4}$ \( 1 + 8 T + 87 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_dj
41$D_{4}$ \( 1 + 8 T + 71 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_ct
43$D_{4}$ \( 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cx
47$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_eo
53$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_fa
61$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_dm
67$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_bq
71$C_2^2$ \( 1 + 139 T^{2} + p^{2} T^{4} \) 2.71.a_fj
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$D_{4}$ \( 1 + 14 T + 159 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_gd
83$D_{4}$ \( 1 - 18 T + 199 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.83.as_hr
89$D_{4}$ \( 1 + 20 T + 266 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.89.u_kg
97$D_{4}$ \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_gg
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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

−7.78782100650750126878789020194, −7.43982672262774386574700286138, −7.20338086686183993768925437584, −7.03802860530981666075432011453, −6.58818915059016548947706123449, −6.39763512303178884412893267255, −5.44608456311610611578839130729, −5.24205119138285064724965158023, −4.94520838389902384741627383162, −4.92498035996240970141285778625, −4.40792131110762427137365325161, −4.12339041481217962380305502922, −3.32020148714545388941363195928, −3.19132175547774889986437545546, −2.78633863692542331696629515097, −2.34035840797413545433604242988, −1.70879534366850979777415105481, −1.44324320674227672351524566687, 0, 0, 1.44324320674227672351524566687, 1.70879534366850979777415105481, 2.34035840797413545433604242988, 2.78633863692542331696629515097, 3.19132175547774889986437545546, 3.32020148714545388941363195928, 4.12339041481217962380305502922, 4.40792131110762427137365325161, 4.92498035996240970141285778625, 4.94520838389902384741627383162, 5.24205119138285064724965158023, 5.44608456311610611578839130729, 6.39763512303178884412893267255, 6.58818915059016548947706123449, 7.03802860530981666075432011453, 7.20338086686183993768925437584, 7.43982672262774386574700286138, 7.78782100650750126878789020194

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