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Properties

Label 4-7098e2-1.1-c1e2-0-20
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
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 + 2·3-s + 3·4-s − 2·5-s + 4·6-s + 2·7-s + 4·8-s + 3·9-s − 4·10-s + 6·12-s + 4·14-s − 4·15-s + 5·16-s − 8·17-s + 6·18-s − 6·19-s − 6·20-s + 4·21-s − 8·23-s + 8·24-s − 4·25-s + 4·27-s + 6·28-s − 14·29-s − 8·30-s − 10·31-s + 6·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s − 1.26·10-s + 1.73·12-s + 1.06·14-s − 1.03·15-s + 5/4·16-s − 1.94·17-s + 1.41·18-s − 1.37·19-s − 1.34·20-s + 0.872·21-s − 1.66·23-s + 1.63·24-s − 4/5·25-s + 0.769·27-s + 1.13·28-s − 2.59·29-s − 1.46·30-s − 1.79·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good5$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_i
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
17$C_2^2$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bv
19$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bj
23$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_by
29$D_{4}$ \( 1 + 14 T + 95 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.29.o_dr
31$D_{4}$ \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_dg
37$D_{4}$ \( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.37.k_ds
41$D_{4}$ \( 1 + 10 T + 95 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_dr
43$D_{4}$ \( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.43.o_fc
47$D_{4}$ \( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_cd
53$D_{4}$ \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_cp
59$D_{4}$ \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_cw
61$D_{4}$ \( 1 - 4 T + 123 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_et
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.67.au_ja
71$D_{4}$ \( 1 + 18 T + 220 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.71.s_im
73$D_{4}$ \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_fi
79$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.79.as_jf
83$D_{4}$ \( 1 - 22 T + 284 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.83.aw_ky
89$D_{4}$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_af
97$D_{4}$ \( 1 - 18 T + 248 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.97.as_jo
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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

−7.72438284331441645994442277819, −7.47503987585896035305854796694, −6.92103059479008605283675970639, −6.85628965048990702360841632767, −6.33846598943127450334182278048, −6.10663717450269031402878192216, −5.33989595158785071607480605357, −5.25951208195818050852881523419, −4.90570268292575245217392590224, −4.30072528008946937512558008115, −4.02534180532585803122346388191, −3.83333684634981055495746208341, −3.54117225658809382692224821416, −3.27809710763417494918608630171, −2.28227700541208189107535460452, −2.10542428974013241310872887259, −1.93597911961949416925105840505, −1.62024100904737084568858725009, 0, 0, 1.62024100904737084568858725009, 1.93597911961949416925105840505, 2.10542428974013241310872887259, 2.28227700541208189107535460452, 3.27809710763417494918608630171, 3.54117225658809382692224821416, 3.83333684634981055495746208341, 4.02534180532585803122346388191, 4.30072528008946937512558008115, 4.90570268292575245217392590224, 5.25951208195818050852881523419, 5.33989595158785071607480605357, 6.10663717450269031402878192216, 6.33846598943127450334182278048, 6.85628965048990702360841632767, 6.92103059479008605283675970639, 7.47503987585896035305854796694, 7.72438284331441645994442277819

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