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Properties

Label 4-7098e2-1.1-c1e2-0-13
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 3·5-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s + 6·10-s − 11-s + 6·12-s − 4·14-s − 6·15-s + 5·16-s − 17-s − 6·18-s − 3·19-s − 9·20-s + 4·21-s + 2·22-s − 7·23-s − 8·24-s + 25-s + 4·27-s + 6·28-s + 29-s + 12·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.34·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 1.89·10-s − 0.301·11-s + 1.73·12-s − 1.06·14-s − 1.54·15-s + 5/4·16-s − 0.242·17-s − 1.41·18-s − 0.688·19-s − 2.01·20-s + 0.872·21-s + 0.426·22-s − 1.45·23-s − 1.63·24-s + 1/5·25-s + 0.769·27-s + 1.13·28-s + 0.185·29-s + 2.19·30-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$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
11$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_s
17$D_{4}$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.17.b_ae
19$D_{4}$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_c
23$D_{4}$ \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.23.h_cc
29$D_{4}$ \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) 2.29.ab_u
31$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_ac
37$D_{4}$ \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.37.al_dw
41$D_{4}$ \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_cw
43$D_{4}$ \( 1 + 17 T + 154 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.43.r_fy
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_cc
59$D_{4}$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_di
61$D_{4}$ \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) 2.61.ab_dg
67$D_{4}$ \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_ew
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$D_{4}$ \( 1 - 3 T + 144 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.73.ad_fo
79$D_{4}$ \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_hi
83$D_{4}$ \( 1 + 26 T + 318 T^{2} + 26 p T^{3} + p^{2} T^{4} \) 2.83.ba_mg
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$D_{4}$ \( 1 - 4 T - 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_acw
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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

−7.892160179088330769831605316233, −7.70948632339476216024453168914, −7.12107149984230423469997921001, −7.07830379124693351335360437209, −6.59166939091016967318563217950, −6.27250817881240225207468998758, −5.60434973517244919085179955025, −5.46791791559510914172237761954, −4.72114635013365308217730092992, −4.34793185877606601802872734577, −4.04726740770731862203467167078, −3.84122513373606083133920354406, −3.10588822765870556371006148251, −2.93744840379310970072257689417, −2.25419486317910645332168112239, −2.15883048094151661725502822760, −1.41426474856606762153725323271, −1.14261155935031524464995188653, 0, 0, 1.14261155935031524464995188653, 1.41426474856606762153725323271, 2.15883048094151661725502822760, 2.25419486317910645332168112239, 2.93744840379310970072257689417, 3.10588822765870556371006148251, 3.84122513373606083133920354406, 4.04726740770731862203467167078, 4.34793185877606601802872734577, 4.72114635013365308217730092992, 5.46791791559510914172237761954, 5.60434973517244919085179955025, 6.27250817881240225207468998758, 6.59166939091016967318563217950, 7.07830379124693351335360437209, 7.12107149984230423469997921001, 7.70948632339476216024453168914, 7.892160179088330769831605316233

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