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Properties

Label 2-39-39.5-c3-0-6
Degree $2$
Conductor $39$
Sign $0.907 + 0.420i$
Analytic cond. $2.30107$
Root an. cond. $1.51692$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.19·3-s − 8i·4-s + (5.58 + 5.58i)7-s + 27·9-s − 41.5i·12-s + (−31.1 + 35i)13-s − 64·16-s + (−105. + 105. i)19-s + (29.0 + 29.0i)21-s − 125i·25-s + 140.·27-s + (44.7 − 44.7i)28-s + (76.0 − 76.0i)31-s − 216i·36-s + (273. + 273. i)37-s + ⋯
L(s)  = 1  + 1.00·3-s − i·4-s + (0.301 + 0.301i)7-s + 9-s − 0.999i·12-s + (−0.665 + 0.746i)13-s − 16-s + (−1.27 + 1.27i)19-s + (0.301 + 0.301i)21-s − i·25-s + 1.00·27-s + (0.301 − 0.301i)28-s + (0.440 − 0.440i)31-s − i·36-s + (1.21 + 1.21i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.907 + 0.420i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.907 + 0.420i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.63622 - 0.360475i\)
\(L(\frac12)\) \(\approx\) \(1.63622 - 0.360475i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19T \)
13 \( 1 + (31.1 - 35i)T \)
good2 \( 1 + 8iT^{2} \)
5 \( 1 + 125iT^{2} \)
7 \( 1 + (-5.58 - 5.58i)T + 343iT^{2} \)
11 \( 1 - 1.33e3iT^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 + (105. - 105. i)T - 6.85e3iT^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-76.0 + 76.0i)T - 2.97e4iT^{2} \)
37 \( 1 + (-273. - 273. i)T + 5.06e4iT^{2} \)
41 \( 1 + 6.89e4iT^{2} \)
43 \( 1 + 218. iT - 7.95e4T^{2} \)
47 \( 1 - 1.03e5iT^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 - 2.05e5iT^{2} \)
61 \( 1 + 935.T + 2.26e5T^{2} \)
67 \( 1 + (-767. + 767. i)T - 3.00e5iT^{2} \)
71 \( 1 + 3.57e5iT^{2} \)
73 \( 1 + (782. + 782. i)T + 3.89e5iT^{2} \)
79 \( 1 - 1.09e3T + 4.93e5T^{2} \)
83 \( 1 + 5.71e5iT^{2} \)
89 \( 1 - 7.04e5iT^{2} \)
97 \( 1 + (1.35e3 - 1.35e3i)T - 9.12e5iT^{2} \)
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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

−15.22573075528765689074228374078, −14.60319707010382095749599833102, −13.65549281705725832436917740111, −12.14147354196192610328973512447, −10.46745627477810550433774939699, −9.444776700306060226029440777901, −8.155419048610548959214006919266, −6.42228761989683509583741006475, −4.49618854745559273726366511349, −2.05501370608218898058971466377, 2.74291168427077067100408965530, 4.38873976500028456531636509264, 7.13323445016229607942645609589, 8.141802739025896144043196910425, 9.309102329239299315269604686839, 10.95147971079917507375939622504, 12.59699299404818048346881928192, 13.33693457882400599878000709476, 14.66661610876046431219491568700, 15.68295580031772008889168602982

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