↑

Properties

Label 2-390-65.29-c1-0-5
Degree $2$
Conductor $390$
Sign $0.841 - 0.540i$
Analytic cond. $3.11416$
Root an. cond. $1.76469$
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  + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (2 + i)5-s + (−0.499 + 0.866i)6-s + (4.33 + 2.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.23 + 0.133i)10-s + (−1.5 − 2.59i)11-s − 0.999i·12-s + (−2.59 + 2.5i)13-s − 5·14-s + (2.23 − 0.133i)15-s + (−0.5 − 0.866i)16-s + (−3.46 − 2i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.894 + 0.447i)5-s + (−0.204 + 0.353i)6-s + (1.63 + 0.944i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.705 + 0.0423i)10-s + (−0.452 − 0.783i)11-s − 0.288i·12-s + (−0.720 + 0.693i)13-s − 1.33·14-s + (0.576 − 0.0345i)15-s + (−0.125 − 0.216i)16-s + (−0.840 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.841 - 0.540i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46355 + 0.429363i\)
\(L(\frac12)\) \(\approx\) \(1.46355 + 0.429363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-2 - i)T \)
13 \( 1 + (2.59 - 2.5i)T \)
good7 \( 1 + (-4.33 - 2.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.46 + 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-7.79 + 4.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5 + 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.3 + 6i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.19 - 3i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6 - 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.66 + 5i)T + (48.5 + 84.0i)T^{2} \)
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

−11.28188679445509902568025139429, −10.43171742012660842586019326638, −9.261542179188313225711077897253, −8.683608509761077420713272736843, −7.82830899626348540891067478516, −6.81366895084444165711807589151, −5.72370042494643021943419985644, −4.80783387313018461080310111111, −2.60553318104467224007780826749, −1.80932312091112183019915576553, 1.45814457036078424350368723308, 2.53687819667944540070053991858, 4.42960344148760180588775099541, 5.01244503294151166544353413361, 6.74976605661738953400008121158, 7.980149495280155477927860625146, 8.296866411723467936506455195064, 9.621669465943787629235875156403, 10.19896027690754945785117392903, 10.93513707150063278531683210937

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