Properties

Label 2-63e2-1.1-c1-0-24
Degree $2$
Conductor $3969$
Sign $1$
Analytic cond. $31.6926$
Root an. cond. $5.62962$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.239·2-s − 1.94·4-s − 2.59·5-s − 0.942·8-s − 0.619·10-s + 4.18·11-s + 3.68·13-s + 3.66·16-s + 1.71·17-s − 7.15·19-s + 5.03·20-s + 0.999·22-s − 5.12·23-s + 1.71·25-s + 0.880·26-s − 2.12·29-s − 6.53·31-s + 2.76·32-s + 0.409·34-s + 1.66·37-s − 1.71·38-s + 2.44·40-s − 10.2·41-s − 1.66·43-s − 8.12·44-s − 1.22·46-s + 9.33·47-s + ⋯
L(s)  = 1  + 0.169·2-s − 0.971·4-s − 1.15·5-s − 0.333·8-s − 0.195·10-s + 1.26·11-s + 1.02·13-s + 0.915·16-s + 0.414·17-s − 1.64·19-s + 1.12·20-s + 0.213·22-s − 1.06·23-s + 0.343·25-s + 0.172·26-s − 0.394·29-s − 1.17·31-s + 0.488·32-s + 0.0701·34-s + 0.272·37-s − 0.277·38-s + 0.386·40-s − 1.59·41-s − 0.253·43-s − 1.22·44-s − 0.180·46-s + 1.36·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.6926\)
Root analytic conductor: \(5.62962\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.018820601\)
\(L(\frac12)\) \(\approx\) \(1.018820601\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 0.239T + 2T^{2} \)
5 \( 1 + 2.59T + 5T^{2} \)
11 \( 1 - 4.18T + 11T^{2} \)
13 \( 1 - 3.68T + 13T^{2} \)
17 \( 1 - 1.71T + 17T^{2} \)
19 \( 1 + 7.15T + 19T^{2} \)
23 \( 1 + 5.12T + 23T^{2} \)
29 \( 1 + 2.12T + 29T^{2} \)
31 \( 1 + 6.53T + 31T^{2} \)
37 \( 1 - 1.66T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 1.66T + 43T^{2} \)
47 \( 1 - 9.33T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 6.06T + 59T^{2} \)
61 \( 1 + 7.98T + 61T^{2} \)
67 \( 1 - 8.26T + 67T^{2} \)
71 \( 1 - 6.23T + 71T^{2} \)
73 \( 1 - 7.15T + 73T^{2} \)
79 \( 1 + 9.82T + 79T^{2} \)
83 \( 1 - 6.89T + 83T^{2} \)
89 \( 1 + 5.03T + 89T^{2} \)
97 \( 1 - 3.06T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.543967245555219207889151656075, −7.924035989780368120033759917079, −6.99744662139277427690464260428, −6.19416750442701545088747473553, −5.46866935232870842959938700318, −4.31987869036341559765755536140, −3.91710892117932815420682975278, −3.49045364410729629812740329400, −1.84702942276003742528666019902, −0.57463188208252772248240942128, 0.57463188208252772248240942128, 1.84702942276003742528666019902, 3.49045364410729629812740329400, 3.91710892117932815420682975278, 4.31987869036341559765755536140, 5.46866935232870842959938700318, 6.19416750442701545088747473553, 6.99744662139277427690464260428, 7.924035989780368120033759917079, 8.543967245555219207889151656075

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