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Properties

Label 4-8379e2-1.1-c1e2-0-7
Degree $4$
Conductor $70207641$
Sign $1$
Analytic cond. $4476.50$
Root an. cond. $8.17964$
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 + 4-s + 4·11-s + 4·13-s + 16-s + 8·17-s + 2·19-s + 8·22-s + 4·23-s − 2·25-s + 8·26-s − 2·32-s + 16·34-s − 12·37-s + 4·38-s + 4·44-s + 8·46-s − 12·47-s − 4·50-s + 4·52-s − 16·53-s + 8·61-s − 11·64-s − 20·67-s + 8·68-s + 20·71-s + 16·73-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 1.20·11-s + 1.10·13-s + 1/4·16-s + 1.94·17-s + 0.458·19-s + 1.70·22-s + 0.834·23-s − 2/5·25-s + 1.56·26-s − 0.353·32-s + 2.74·34-s − 1.97·37-s + 0.648·38-s + 0.603·44-s + 1.17·46-s − 1.75·47-s − 0.565·50-s + 0.554·52-s − 2.19·53-s + 1.02·61-s − 1.37·64-s − 2.44·67-s + 0.970·68-s + 2.37·71-s + 1.87·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70207641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70207641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(70207641\)    =    \(3^{4} \cdot 7^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(4476.50\)
Root analytic conductor: \(8.17964\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 70207641,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.195262395\)
\(L(\frac12)\) \(\approx\) \(7.195262395\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
7 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_d
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_s
13$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_w
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bq
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bq
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.29.a_ao
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.41.a_by
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_es
53$D_{4}$ \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.53.q_gg
59$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.59.a_di
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$D_{4}$ \( 1 + 20 T + 226 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.67.u_is
71$D_{4}$ \( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.71.au_ic
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.73.aq_ic
79$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_es
83$D_{4}$ \( 1 - 4 T + 162 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_gg
89$D_{4}$ \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.89.y_le
97$D_{4}$ \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_gc
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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

−8.078995270448619306687142311721, −7.40485734412077937150370935600, −7.33886903981024334959935896958, −6.63540691769122829687294287381, −6.59304639618179987489327212679, −6.25991864105311545734733336830, −5.67750786425896670368784004197, −5.44614701128229507094230519824, −5.12468948513043959496554063908, −5.02894034070982118769634141626, −4.22377232870271963348566590360, −4.19310174736509992724304221170, −3.57688474722873815476678187952, −3.57453954600856998698479638833, −2.98968921574449153862325786165, −2.92176518433920658930538602942, −1.82179982687323108952470841901, −1.42359303023809217045226800641, −1.38551194748689730126551863629, −0.49326775518022342577598690799, 0.49326775518022342577598690799, 1.38551194748689730126551863629, 1.42359303023809217045226800641, 1.82179982687323108952470841901, 2.92176518433920658930538602942, 2.98968921574449153862325786165, 3.57453954600856998698479638833, 3.57688474722873815476678187952, 4.19310174736509992724304221170, 4.22377232870271963348566590360, 5.02894034070982118769634141626, 5.12468948513043959496554063908, 5.44614701128229507094230519824, 5.67750786425896670368784004197, 6.25991864105311545734733336830, 6.59304639618179987489327212679, 6.63540691769122829687294287381, 7.33886903981024334959935896958, 7.40485734412077937150370935600, 8.078995270448619306687142311721

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