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Properties

Label 2-3639-3639.3638-c0-0-3
Degree $2$
Conductor $3639$
Sign $1$
Analytic cond. $1.81609$
Root an. cond. $1.34762$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 3-s + 1.61·4-s − 1.61·5-s − 1.61·6-s − 1.61·7-s − 8-s + 9-s + 2.61·10-s + 1.61·12-s − 1.61·13-s + 2.61·14-s − 1.61·15-s − 1.61·17-s − 1.61·18-s − 1.61·19-s − 2.61·20-s − 1.61·21-s + 0.618·23-s − 24-s + 1.61·25-s + 2.61·26-s + 27-s − 2.61·28-s + 2.61·30-s + 0.618·31-s + 32-s + ⋯
L(s)  = 1  − 1.61·2-s + 3-s + 1.61·4-s − 1.61·5-s − 1.61·6-s − 1.61·7-s − 8-s + 9-s + 2.61·10-s + 1.61·12-s − 1.61·13-s + 2.61·14-s − 1.61·15-s − 1.61·17-s − 1.61·18-s − 1.61·19-s − 2.61·20-s − 1.61·21-s + 0.618·23-s − 24-s + 1.61·25-s + 2.61·26-s + 27-s − 2.61·28-s + 2.61·30-s + 0.618·31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3639\)    =    \(3 \cdot 1213\)
Sign: $1$
Analytic conductor: \(1.81609\)
Root analytic conductor: \(1.34762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3639} (3638, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3639,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2832306896\)
\(L(\frac12)\) \(\approx\) \(0.2832306896\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
1213 \( 1 - T \)
good2 \( 1 + 1.61T + T^{2} \)
5 \( 1 + 1.61T + T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.61T + T^{2} \)
17 \( 1 + 1.61T + T^{2} \)
19 \( 1 + 1.61T + T^{2} \)
23 \( 1 - 0.618T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 - 0.618T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 0.618T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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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

−8.780114460318962019851392207652, −8.141100445123422672585010581866, −7.30982590235970481314265417335, −7.08676987828406579057372195616, −6.35912001843893550732570435027, −4.41436039403187068313324607922, −4.08992264214386562979669047519, −2.71698492704198616479275810868, −2.45015255854962338853240151454, −0.51504337609766155587676597342, 0.51504337609766155587676597342, 2.45015255854962338853240151454, 2.71698492704198616479275810868, 4.08992264214386562979669047519, 4.41436039403187068313324607922, 6.35912001843893550732570435027, 7.08676987828406579057372195616, 7.30982590235970481314265417335, 8.141100445123422672585010581866, 8.780114460318962019851392207652

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