↑

Properties

Label 2-39-13.8-c2-0-3
Degree $2$
Conductor $39$
Sign $-0.702 + 0.711i$
Analytic cond. $1.06267$
Root an. cond. $1.03086$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.26 − 1.26i)2-s − 1.73·3-s − 0.798i·4-s + (−4.65 − 4.65i)5-s + (2.19 + 2.19i)6-s + (1.87 − 1.87i)7-s + (−6.07 + 6.07i)8-s + 2.99·9-s + 11.7i·10-s + (11.0 − 11.0i)11-s + 1.38i·12-s + (11.4 + 6.06i)13-s − 4.75·14-s + (8.06 + 8.06i)15-s + 12.1·16-s + 3.20i·17-s + ⋯
L(s)  = 1  + (−0.632 − 0.632i)2-s − 0.577·3-s − 0.199i·4-s + (−0.931 − 0.931i)5-s + (0.365 + 0.365i)6-s + (0.268 − 0.268i)7-s + (−0.758 + 0.758i)8-s + 0.333·9-s + 1.17i·10-s + (1.00 − 1.00i)11-s + 0.115i·12-s + (0.884 + 0.466i)13-s − 0.339·14-s + (0.537 + 0.537i)15-s + 0.760·16-s + 0.188i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.702 + 0.711i$
Analytic conductor: \(1.06267\)
Root analytic conductor: \(1.03086\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1),\ -0.702 + 0.711i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.212229 - 0.508118i\)
\(L(\frac12)\) \(\approx\) \(0.212229 - 0.508118i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
13 \( 1 + (-11.4 - 6.06i)T \)
good2 \( 1 + (1.26 + 1.26i)T + 4iT^{2} \)
5 \( 1 + (4.65 + 4.65i)T + 25iT^{2} \)
7 \( 1 + (-1.87 + 1.87i)T - 49iT^{2} \)
11 \( 1 + (-11.0 + 11.0i)T - 121iT^{2} \)
17 \( 1 - 3.20iT - 289T^{2} \)
19 \( 1 + (19.7 + 19.7i)T + 361iT^{2} \)
23 \( 1 - 3.84iT - 529T^{2} \)
29 \( 1 - 25.4T + 841T^{2} \)
31 \( 1 + (25.7 + 25.7i)T + 961iT^{2} \)
37 \( 1 + (-35.2 + 35.2i)T - 1.36e3iT^{2} \)
41 \( 1 + (-41.0 - 41.0i)T + 1.68e3iT^{2} \)
43 \( 1 - 51.1iT - 1.84e3T^{2} \)
47 \( 1 + (-0.856 + 0.856i)T - 2.20e3iT^{2} \)
53 \( 1 + 39.2T + 2.80e3T^{2} \)
59 \( 1 + (-54.7 + 54.7i)T - 3.48e3iT^{2} \)
61 \( 1 - 50.5T + 3.72e3T^{2} \)
67 \( 1 + (1.32 + 1.32i)T + 4.48e3iT^{2} \)
71 \( 1 + (10.8 + 10.8i)T + 5.04e3iT^{2} \)
73 \( 1 + (91.2 - 91.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 95.2T + 6.24e3T^{2} \)
83 \( 1 + (20.5 + 20.5i)T + 6.88e3iT^{2} \)
89 \( 1 + (71.2 - 71.2i)T - 7.92e3iT^{2} \)
97 \( 1 + (-28.4 - 28.4i)T + 9.40e3iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.90229601459738281437847430901, −14.46300071822782809327746035646, −12.85100623723419785205862558093, −11.39456648775723331396706858319, −11.14493292080846773750196858265, −9.268253990503975919829497548866, −8.319725717180099591951039635340, −6.16912531339483720909952502734, −4.30207349851120949249536880178, −0.893582822051883630884892976829, 3.86473272019059703044966353948, 6.37323303951126276055559704336, 7.38752350060111876012239379157, 8.647719334869752701665622913581, 10.34146852372127981291283636087, 11.65945856964439467475927562875, 12.55468169466272420335860685679, 14.65231727225159388066840835163, 15.45177608089529926143625556347, 16.48337310709179536544459443650

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