Properties

Label 2-63e2-1.1-c1-0-6
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  − 1.69·2-s + 0.888·4-s − 0.949·5-s + 1.88·8-s + 1.61·10-s − 0.588·11-s − 5.01·13-s − 4.98·16-s − 7.58·17-s − 4.46·19-s − 0.843·20-s + 22-s + 2.47·23-s − 4.09·25-s + 8.53·26-s + 5.47·29-s − 6.07·31-s + 4.69·32-s + 12.8·34-s − 6.98·37-s + 7.58·38-s − 1.79·40-s − 1.05·41-s + 6.98·43-s − 0.522·44-s − 4.21·46-s − 7.47·47-s + ⋯
L(s)  = 1  − 1.20·2-s + 0.444·4-s − 0.424·5-s + 0.667·8-s + 0.510·10-s − 0.177·11-s − 1.39·13-s − 1.24·16-s − 1.83·17-s − 1.02·19-s − 0.188·20-s + 0.213·22-s + 0.516·23-s − 0.819·25-s + 1.67·26-s + 1.01·29-s − 1.09·31-s + 0.830·32-s + 2.21·34-s − 1.14·37-s + 1.23·38-s − 0.283·40-s − 0.164·41-s + 1.06·43-s − 0.0788·44-s − 0.620·46-s − 1.09·47-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

\(L(1)\) \(\approx\) \(0.2929071142\)
\(L(\frac12)\) \(\approx\) \(0.2929071142\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.69T + 2T^{2} \)
5 \( 1 + 0.949T + 5T^{2} \)
11 \( 1 + 0.588T + 11T^{2} \)
13 \( 1 + 5.01T + 13T^{2} \)
17 \( 1 + 7.58T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 - 5.47T + 29T^{2} \)
31 \( 1 + 6.07T + 31T^{2} \)
37 \( 1 + 6.98T + 37T^{2} \)
41 \( 1 + 1.05T + 41T^{2} \)
43 \( 1 - 6.98T + 43T^{2} \)
47 \( 1 + 7.47T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 - 4.30T + 71T^{2} \)
73 \( 1 - 4.46T + 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 + 5.68T + 83T^{2} \)
89 \( 1 - 0.843T + 89T^{2} \)
97 \( 1 + 3.40T + 97T^{2} \)
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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

−8.525115479879567909665487699490, −7.930754246897346903326059484425, −6.96994119216770458668787820567, −6.83274156529634052410936515538, −5.42432966229033177441234855867, −4.59246501100130431339647149665, −4.00458516795650871789295242193, −2.55123928727730680987026449706, −1.89422952642985197485481230319, −0.35615733911652037239925243447, 0.35615733911652037239925243447, 1.89422952642985197485481230319, 2.55123928727730680987026449706, 4.00458516795650871789295242193, 4.59246501100130431339647149665, 5.42432966229033177441234855867, 6.83274156529634052410936515538, 6.96994119216770458668787820567, 7.930754246897346903326059484425, 8.525115479879567909665487699490

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