Properties

Label 2-63e2-1.1-c1-0-84
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  − 1.53·2-s + 0.360·4-s + 3.15·5-s + 2.51·8-s − 4.85·10-s + 5.74·11-s + 0.360·13-s − 4.59·16-s + 2.77·17-s + 7.23·19-s + 1.13·20-s − 8.83·22-s + 0.824·23-s + 4.97·25-s − 0.554·26-s + 4.27·29-s + 4.98·31-s + 2.01·32-s − 4.26·34-s + 7.49·37-s − 11.1·38-s + 7.95·40-s − 3.33·41-s − 7.86·43-s + 2.07·44-s − 1.26·46-s − 3.48·47-s + ⋯
L(s)  = 1  − 1.08·2-s + 0.180·4-s + 1.41·5-s + 0.890·8-s − 1.53·10-s + 1.73·11-s + 0.100·13-s − 1.14·16-s + 0.673·17-s + 1.65·19-s + 0.254·20-s − 1.88·22-s + 0.171·23-s + 0.994·25-s − 0.108·26-s + 0.794·29-s + 0.894·31-s + 0.356·32-s − 0.731·34-s + 1.23·37-s − 1.80·38-s + 1.25·40-s − 0.520·41-s − 1.20·43-s + 0.312·44-s − 0.186·46-s − 0.507·47-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

\(L(1)\) \(\approx\) \(1.774352508\)
\(L(\frac12)\) \(\approx\) \(1.774352508\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.53T + 2T^{2} \)
5 \( 1 - 3.15T + 5T^{2} \)
11 \( 1 - 5.74T + 11T^{2} \)
13 \( 1 - 0.360T + 13T^{2} \)
17 \( 1 - 2.77T + 17T^{2} \)
19 \( 1 - 7.23T + 19T^{2} \)
23 \( 1 - 0.824T + 23T^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 - 4.98T + 31T^{2} \)
37 \( 1 - 7.49T + 37T^{2} \)
41 \( 1 + 3.33T + 41T^{2} \)
43 \( 1 + 7.86T + 43T^{2} \)
47 \( 1 + 3.48T + 47T^{2} \)
53 \( 1 - 2.91T + 53T^{2} \)
59 \( 1 - 2.39T + 59T^{2} \)
61 \( 1 + 3.20T + 61T^{2} \)
67 \( 1 - 1.89T + 67T^{2} \)
71 \( 1 + 1.60T + 71T^{2} \)
73 \( 1 - 15.4T + 73T^{2} \)
79 \( 1 - 5.47T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−8.607841460881679186299435000034, −7.950412597388767987837210143703, −6.94610348365603553891516947973, −6.47179518133398942922048421530, −5.57563035463830511352757176983, −4.82382093248910994549807705560, −3.79071963029528475234662684786, −2.69671159140971554176645505054, −1.43650366893265572994786448288, −1.11082190766082195375261712101, 1.11082190766082195375261712101, 1.43650366893265572994786448288, 2.69671159140971554176645505054, 3.79071963029528475234662684786, 4.82382093248910994549807705560, 5.57563035463830511352757176983, 6.47179518133398942922048421530, 6.94610348365603553891516947973, 7.950412597388767987837210143703, 8.607841460881679186299435000034

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