Properties

Label 2-39-1.1-c3-0-2
Degree $2$
Conductor $39$
Sign $1$
Analytic cond. $2.30107$
Root an. cond. $1.51692$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.52·2-s + 3·3-s − 5.66·4-s + 19.3·5-s + 4.57·6-s + 4.84·7-s − 20.8·8-s + 9·9-s + 29.5·10-s − 61.0·11-s − 17.0·12-s + 13·13-s + 7.39·14-s + 58.0·15-s + 13.5·16-s − 41.7·17-s + 13.7·18-s − 107.·19-s − 109.·20-s + 14.5·21-s − 93.2·22-s + 28.5·23-s − 62.5·24-s + 249.·25-s + 19.8·26-s + 27·27-s − 27.4·28-s + ⋯
L(s)  = 1  + 0.539·2-s + 0.577·3-s − 0.708·4-s + 1.72·5-s + 0.311·6-s + 0.261·7-s − 0.922·8-s + 0.333·9-s + 0.933·10-s − 1.67·11-s − 0.409·12-s + 0.277·13-s + 0.141·14-s + 0.998·15-s + 0.211·16-s − 0.596·17-s + 0.179·18-s − 1.29·19-s − 1.22·20-s + 0.150·21-s − 0.903·22-s + 0.258·23-s − 0.532·24-s + 1.99·25-s + 0.149·26-s + 0.192·27-s − 0.185·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $1$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.906770496\)
\(L(\frac12)\) \(\approx\) \(1.906770496\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 - 13T \)
good2 \( 1 - 1.52T + 8T^{2} \)
5 \( 1 - 19.3T + 125T^{2} \)
7 \( 1 - 4.84T + 343T^{2} \)
11 \( 1 + 61.0T + 1.33e3T^{2} \)
17 \( 1 + 41.7T + 4.91e3T^{2} \)
19 \( 1 + 107.T + 6.85e3T^{2} \)
23 \( 1 - 28.5T + 1.21e4T^{2} \)
29 \( 1 + 89.8T + 2.43e4T^{2} \)
31 \( 1 - 183.T + 2.97e4T^{2} \)
37 \( 1 - 418.T + 5.06e4T^{2} \)
41 \( 1 + 142.T + 6.89e4T^{2} \)
43 \( 1 + 71.0T + 7.95e4T^{2} \)
47 \( 1 - 323.T + 1.03e5T^{2} \)
53 \( 1 + 25.1T + 1.48e5T^{2} \)
59 \( 1 + 684.T + 2.05e5T^{2} \)
61 \( 1 - 308.T + 2.26e5T^{2} \)
67 \( 1 - 672.T + 3.00e5T^{2} \)
71 \( 1 + 326.T + 3.57e5T^{2} \)
73 \( 1 - 24.3T + 3.89e5T^{2} \)
79 \( 1 - 166.T + 4.93e5T^{2} \)
83 \( 1 + 201.T + 5.71e5T^{2} \)
89 \( 1 - 108.T + 7.04e5T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−15.34223632232878242393670478846, −14.30044286038327071267886798589, −13.25708409642756462427154423655, −13.00245693476129854720288499989, −10.57736556386613563650598886951, −9.502178898180561564865884871245, −8.329304548051192620647923242570, −6.09547993294305422456410394519, −4.81817224080894136526122061989, −2.50142958496538768405667615354, 2.50142958496538768405667615354, 4.81817224080894136526122061989, 6.09547993294305422456410394519, 8.329304548051192620647923242570, 9.502178898180561564865884871245, 10.57736556386613563650598886951, 13.00245693476129854720288499989, 13.25708409642756462427154423655, 14.30044286038327071267886798589, 15.34223632232878242393670478846

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