↑

Properties

Label 2-39-13.12-c1-0-0
Degree $2$
Conductor $39$
Sign $0.277 - 0.960i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·2-s − 3-s − 0.999·4-s − 1.73i·6-s − 3.46i·7-s + 1.73i·8-s + 9-s − 3.46i·11-s + 0.999·12-s + (−1 + 3.46i)13-s + 5.99·14-s − 5·16-s − 6·17-s + 1.73i·18-s + 3.46i·19-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.577·3-s − 0.499·4-s − 0.707i·6-s − 1.30i·7-s + 0.612i·8-s + 0.333·9-s − 1.04i·11-s + 0.288·12-s + (−0.277 + 0.960i)13-s + 1.60·14-s − 1.25·16-s − 1.45·17-s + 0.408i·18-s + 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.277 - 0.960i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550763 + 0.414261i\)
\(L(\frac12)\) \(\approx\) \(0.550763 + 0.414261i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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

−16.68934332216539481158586929182, −15.66671800979046012428842867774, −14.26255209441121924621344692766, −13.46495961562743261746007621118, −11.58516389668836280292342537270, −10.53459028746773639400487486073, −8.644174139697889043440842449786, −7.17583948104702118314526429719, −6.27213339115059087211963100445, −4.53114121658275230059828339895, 2.50907115140597324194211607825, 4.86905381270888678414367702714, 6.74370882529777620950526970910, 8.940367594292810735571675595475, 10.21653240500991125494313664343, 11.32665602148099343374542466665, 12.34089512118979564148306677207, 13.00091491120810677917759627526, 15.10589552252558216865316324167, 15.86429046282199755109551449657

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