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Properties

Label 2-39-13.10-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: $\chi_{39} (10, \cdot )$
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

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

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