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Properties

Label 2-39-39.2-c1-0-1
Degree $2$
Conductor $39$
Sign $0.953 + 0.302i$
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  + (−0.866 − 1.5i)3-s + (1.73 + i)4-s + (−4.23 + 1.13i)7-s + (−1.5 + 2.59i)9-s − 3.46i·12-s + (2.59 − 2.5i)13-s + (1.99 + 3.46i)16-s + (−0.830 − 3.09i)19-s + (5.36 + 5.36i)21-s − 5i·25-s + 5.19·27-s + (−8.46 − 2.26i)28-s + (0.830 − 0.830i)31-s + (−5.19 + 3i)36-s + (−3.09 + 11.5i)37-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (0.866 + 0.5i)4-s + (−1.59 + 0.428i)7-s + (−0.5 + 0.866i)9-s − 0.999i·12-s + (0.720 − 0.693i)13-s + (0.499 + 0.866i)16-s + (−0.190 − 0.710i)19-s + (1.17 + 1.17i)21-s − i·25-s + 1.00·27-s + (−1.59 − 0.428i)28-s + (0.149 − 0.149i)31-s + (−0.866 + 0.5i)36-s + (−0.509 + 1.90i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.953 + 0.302i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.953 + 0.302i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.706662 - 0.109269i\)
\(L(\frac12)\) \(\approx\) \(0.706662 - 0.109269i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
13 \( 1 + (-2.59 + 2.5i)T \)
good2 \( 1 + (-1.73 - i)T^{2} \)
5 \( 1 + 5iT^{2} \)
7 \( 1 + (4.23 - 1.13i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.830 + 3.09i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.830 + 0.830i)T - 31iT^{2} \)
37 \( 1 + (3.09 - 11.5i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.33 - 7.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-15.7 - 4.23i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (7.63 + 7.63i)T + 73iT^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.57 + 9.59i)T + (-84.0 + 48.5i)T^{2} \)
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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

−16.24644086000657269718930235816, −15.47068144434049292699101142043, −13.40704597303938166863939693732, −12.65208779821095528365292614123, −11.68233344115948147884910941145, −10.33799096166553589972307571081, −8.407971758694125939444127237067, −6.87588616744967647746205702468, −6.04231259098432226043157725372, −2.90998648777473488123802691863, 3.59850093997033233524973792826, 5.83187567293754489209147142198, 6.84682932914977747208997913860, 9.311099632916972730542005383458, 10.28492346517608797924035033166, 11.27827400375444776803415132348, 12.62207203507745843343992234168, 14.23104170099924442255296359959, 15.59384536352613973508526326816, 16.19415011714200628062315027242

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