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Properties

Label 4-1040e2-1.1-c1e2-0-47
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 2·5-s + 8·9-s + 8·11-s − 6·13-s − 8·15-s + 2·17-s + 8·23-s − 25-s + 12·27-s + 4·31-s + 32·33-s − 24·39-s − 14·41-s − 4·43-s − 16·45-s − 2·49-s + 8·51-s + 2·53-s − 16·55-s + 16·61-s + 12·65-s + 24·67-s + 32·69-s − 4·71-s − 24·73-s − 4·75-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s + 8/3·9-s + 2.41·11-s − 1.66·13-s − 2.06·15-s + 0.485·17-s + 1.66·23-s − 1/5·25-s + 2.30·27-s + 0.718·31-s + 5.57·33-s − 3.84·39-s − 2.18·41-s − 0.609·43-s − 2.38·45-s − 2/7·49-s + 1.12·51-s + 0.274·53-s − 2.15·55-s + 2.04·61-s + 1.48·65-s + 2.93·67-s + 3.85·69-s − 0.474·71-s − 2.80·73-s − 0.461·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.793718406\)
\(L(\frac12)\) \(\approx\) \(4.793718406\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.3.ae_i
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.11.ai_bg
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_c
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
23$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bg
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.29.a_acc
31$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_i
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.41.o_du
43$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_i
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_c
59$C_2^2$ \( 1 + p^{2} T^{4} \) 2.59.a_a
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_i
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.73.y_le
79$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.79.a_du
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.89.ao_du
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−10.06327715423773923856580581578, −9.538145543295807194478994515769, −9.253552527090642176285385961112, −8.747011854794860601993851006262, −8.600216482669408025491713822201, −8.146668043197777533724535625927, −7.83808274947745396198671334171, −7.15067070097182128293684984779, −6.88883705888303204295125845309, −6.84776779776205515915804275821, −5.98830575258327547785253502918, −5.06432232159111346475109885812, −4.82882043890564966235748970245, −4.12233343329807749326428666071, −3.80367551138514115651417984369, −3.26537912217688673109113933405, −3.06046520907601084162166633298, −2.30368110465444933339864586058, −1.75353872416620285862189438591, −0.915754750883191810465704416978, 0.915754750883191810465704416978, 1.75353872416620285862189438591, 2.30368110465444933339864586058, 3.06046520907601084162166633298, 3.26537912217688673109113933405, 3.80367551138514115651417984369, 4.12233343329807749326428666071, 4.82882043890564966235748970245, 5.06432232159111346475109885812, 5.98830575258327547785253502918, 6.84776779776205515915804275821, 6.88883705888303204295125845309, 7.15067070097182128293684984779, 7.83808274947745396198671334171, 8.146668043197777533724535625927, 8.600216482669408025491713822201, 8.747011854794860601993851006262, 9.253552527090642176285385961112, 9.538145543295807194478994515769, 10.06327715423773923856580581578

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