Properties

Label 4-980e2-1.1-c1e2-0-15
Degree $4$
Conductor $960400$
Sign $1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
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  − 3·3-s − 5-s + 3·9-s + 2·11-s + 12·13-s + 3·15-s + 2·17-s + 9·23-s + 6·29-s + 2·31-s − 6·33-s − 8·37-s − 36·39-s − 10·41-s + 2·43-s − 3·45-s + 8·47-s − 6·51-s − 4·53-s − 2·55-s − 8·59-s + 7·61-s − 12·65-s + 3·67-s − 27·69-s + 16·71-s + 14·73-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 9-s + 0.603·11-s + 3.32·13-s + 0.774·15-s + 0.485·17-s + 1.87·23-s + 1.11·29-s + 0.359·31-s − 1.04·33-s − 1.31·37-s − 5.76·39-s − 1.56·41-s + 0.304·43-s − 0.447·45-s + 1.16·47-s − 0.840·51-s − 0.549·53-s − 0.269·55-s − 1.04·59-s + 0.896·61-s − 1.48·65-s + 0.366·67-s − 3.25·69-s + 1.89·71-s + 1.63·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{980} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.402513147\)
\(L(\frac12)\) \(\approx\) \(1.402513147\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−10.31129931620928935996612885302, −10.12415045780528937101596296056, −9.140461885245802917817875130940, −9.019429307288767411934948838560, −8.575985217166682750136144618897, −8.199973389632290018009885314443, −7.74886919339459238350470747174, −6.89064595664227616231617602403, −6.69752257954642342614077714698, −6.40432213967838869900548633656, −5.83866256231900707754218541761, −5.66851783944335284476522793276, −4.95869957540646618009965506755, −4.74011430270501850512274903563, −3.90327264761187513524774144251, −3.35323353790531445569869501298, −3.35284352103996439862655159452, −1.96222052862298600103783583546, −0.998071215017101942980472863605, −0.869566947810801854820645664028, 0.869566947810801854820645664028, 0.998071215017101942980472863605, 1.96222052862298600103783583546, 3.35284352103996439862655159452, 3.35323353790531445569869501298, 3.90327264761187513524774144251, 4.74011430270501850512274903563, 4.95869957540646618009965506755, 5.66851783944335284476522793276, 5.83866256231900707754218541761, 6.40432213967838869900548633656, 6.69752257954642342614077714698, 6.89064595664227616231617602403, 7.74886919339459238350470747174, 8.199973389632290018009885314443, 8.575985217166682750136144618897, 9.019429307288767411934948838560, 9.140461885245802917817875130940, 10.12415045780528937101596296056, 10.31129931620928935996612885302

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