Properties

Label 2-39-13.4-c1-0-0
Degree $2$
Conductor $39$
Sign $0.964 + 0.265i$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−1 − 1.73i)4-s + 3.46i·5-s + (−1.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−3 − 1.73i)11-s − 1.99·12-s + (3.5 − 0.866i)13-s + (2.99 + 1.73i)15-s + (−1.99 + 3.46i)16-s + (3 − 1.73i)19-s + (5.99 − 3.46i)20-s + 1.73i·21-s + (3 − 5.19i)23-s − 6.99·25-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.5 − 0.866i)4-s + 1.54i·5-s + (−0.566 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.904 − 0.522i)11-s − 0.577·12-s + (0.970 − 0.240i)13-s + (0.774 + 0.447i)15-s + (−0.499 + 0.866i)16-s + (0.688 − 0.397i)19-s + (1.34 − 0.774i)20-s + 0.377i·21-s + (0.625 − 1.08i)23-s − 1.39·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.729602 - 0.0984346i\)
\(L(\frac12)\) \(\approx\) \(0.729602 - 0.0984346i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good2 \( 1 + (1 + 1.73i)T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 + (1.5 - 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6 + 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (3 - 1.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9 - 5.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (-6 - 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−15.88808616152548985528097002510, −14.91846109816130859091273293097, −13.96419287562541034555919049480, −13.04619436231722837529409926035, −11.10551651873457236367808656979, −10.26712380940950273475966635755, −8.739124385087970338720574082298, −6.96894414488669299222772554564, −5.79122345317736035221054158129, −3.03451756138043372419512126667, 3.79028449147413468164063210595, 5.16881605044338984679923592521, 7.75582342999115831642782779487, 8.848690190067500122946250492219, 9.818472480147079412536725966363, 11.78493119263088006316411982323, 13.11314330705065684621775478908, 13.47313273791666928233950183414, 15.55977393109343258918665426919, 16.40034926286238632389372103235

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